{"group":{"id":1,"name":"Community","lockable":false,"created_at":"2012-01-18T18:02:15.000Z","updated_at":"2026-04-06T14:01:22.000Z","description":"Problems submitted by members of the MATLAB Central community.","is_default":true,"created_by":161519,"badge_id":null,"featured":false,"trending":false,"solution_count_in_trending_period":0,"trending_last_calculated":"2026-04-06T00:00:00.000Z","image_id":null,"published":true,"community_created":false,"status_id":2,"is_default_group_for_player":false,"deleted_by":null,"deleted_at":null,"restored_by":null,"restored_at":null,"description_opc":null,"description_html":null,"published_at":null},"problems":[{"id":61068,"title":"Average Number of Moves for the Royal Game of Err","description":"Bored of tedious court assemblies, King Neduchadneddar the Procrastinator has found a cunning way to keep his advisors, councilors, viziers, and other courtly lackeys busy: he plays a simple and pointless board game with them.\r\nPlayers move their pawn along a path of n squares, starting at square 1, with the goal of being the first to reach square n. On each turn, they roll an m-sided die to determine how many squares to move. However, if their move would take them beyond the nth square, they stay put. Also, players cannot land on any of the numbers that King Neduchadneddar has decided are the unlucky numbers of the day. If their move would land them on any of these squares, they stay put instead.\r\nTo keep his courtiers on their toes (so they have less time to plot his downfall), the king changes the number of squares (n), the number of sides of the die (m), and the list of unlucky forbidden squares each day.\r\nNeeding to know how much time he will waste with each game, the king in his wisdom (and/or insanity) has appointed you Chief Royal Mage of Matrix Computations and tasked you with determining the average number of moves it will take a player to complete a game according to today’s setup.\r\nYou can use the transition matrix for the game to do this. The transition matrix T is an n-by-n matrix where T(j,k) is the probability of moving from square j to square k in one move. Then  gives the probability of moving from from square j to square k in m moves (where  is a matrix power; that is, , using standard matrix multiplication).\r\n is the probability of moving from square 1 to square n in m moves. That is, it is the probability of completing the game in m moves. However, once you reach square n you stay there forever. (In practice, the game would end at this point.) Therefore,  increases monotonically towards 1 as m increases, and can be interpreted as the cumulative probability of completing the game in m moves. Or, in other words, the probability of completing the game in m moves or fewer.\r\nThis is convenient because there's a result in probability theory that allows you to calculate the expected value of m (that is, the average number of moves to complete the game) from the cumulative probability. Skipping the math details, the punchline is:\r\n\r\nAn infinite sum may not seem helpful, but remember that  increases monotonically towards 1, which means the terms in the sum go to zero. Therefore, in practice, you can find the average number of moves to complete the game by summing  until the terms are too small to make any practical difference.\r\nGiven a transition matrix T, return the average number of moves , accurate to 3 decimal places.\r\nExample\r\nWith n = 8 and m = 3, and square 5 forbidden, the transition matrix is:\r\nT = [0      1/3    1/3    1/3    0      0      0      0       \r\n    0      1/3    1/3    1/3    0      0      0      0       \r\n    0      0      1/3    1/3    0      1/3    0      0       \r\n    0      0      0      1/3    0      1/3    1/3    0       \r\n    0      0      0      0      0      0      0      0       \r\n    0      0      0      0      0      1/3    1/3    1/3     \r\n    0      0      0      0      0      0      2/3    1/3     \r\n    0      0      0      0      0      0      0      1];\r\n\r\navgm = avgmoves(T)\r\navgm =\r\n6.3750\r\nAn inexact way to verify the result is to directly simulate playing the game many times:\r\nfunction avgm = simulategame(n,m,nogo,nexp)\r\nmv = zeros(nexp,1);\r\nfor g = 1:nexp\r\n    % Start game at 1\r\n    s = 1;\r\n    % Go until we reach the nth square\r\n    while (s \u003c n)\r\n        % Roll die and see where we would land\r\n        snext = s + randi(m);\r\n        if (snext \u003c= n) \u0026\u0026 ~any(snext == nogo)\r\n            % Move only if valid\r\n            s = snext;\r\n        end\r\n        % Increment number of moves for this game\r\n        mv(g) = mv(g) + 1;\r\n    end\r\n    % Go to the next game\r\nend\r\navgm = mean(mv);\r\nend\r\n\r\nn = 8;\r\nm = 3;\r\nnogo = 5;\r\navgmsim = simulategame(n,m,nogo,1e4)\r\navgmsim =\r\n6.3626\r\nThis approach would require too many simulations to be accurate to the required precision, but gives rough verification that the game takes 6.375 moves on average.\r\nFor n = 16, m = 5, and the forbidden squares are 4, 6, 10, 11, and 15, the transition matrix is:\r\nidx = [1;17;18;33;34;35;65;66;67;69;98;99;101;103;115;117;119;120;133;135;136;137;183;184;185;188;200;201;204;205;217;220;221;222;252;253;254;256];\r\nTvals = [0.4;0.2;0.4;0.2;0.2;0.4;0.2;0.2;0.2;0.4;0.2;0.2;0.2;0.4;0.2;0.2;0.2;0.4;0.2;0.2;0.2;0.4;0.2;0.2;0.2;0.4;0.2;0.2;0.2;0.6;0.2;0.2;0.2;0.8;0.2;0.2;0.2;1];\r\nT = zeros(16);\r\nT(idx) = Tvals\r\nT =\r\n0.4000    0.2000    0.2000         0    0.2000         0         0         0         0         0         0         0         0         0         0         0\r\n0    0.4000    0.2000         0    0.2000         0    0.2000         0         0         0         0         0         0         0         0         0\r\n0         0    0.4000         0    0.2000         0    0.2000    0.2000         0         0         0         0         0         0         0         0\r\n0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0\r\n0         0         0         0    0.4000         0    0.2000    0.2000    0.2000         0         0         0         0         0         0         0\r\n0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0\r\n0         0         0         0         0         0    0.4000    0.2000    0.2000         0         0    0.2000         0         0         0         0\r\n0         0         0         0         0         0         0    0.4000    0.2000         0         0    0.2000    0.2000         0         0         0\r\n0         0         0         0         0         0         0         0    0.4000         0         0    0.2000    0.2000    0.2000         0         0\r\n0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0\r\n0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0\r\n0         0         0         0         0         0         0         0         0         0         0    0.4000    0.2000    0.2000         0    0.2000\r\n0         0         0         0         0         0         0         0         0         0         0         0    0.6000    0.2000         0    0.2000\r\n0         0         0         0         0         0         0         0         0         0         0         0         0    0.8000         0    0.2000\r\n0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0\r\n0         0         0         0         0         0         0         0         0         0         0         0         0         0         0    1.0000\r\nComputing the average number of moves from the transition matrix and from simulation:\r\navgm = avgmoves(T)\r\navgm =\r\n11.4015\r\navgmsim = simulategame(16,5,[4 6 10 11 15],1e4)\r\navgmsim =\r\n11.4266\r\nAssumptions\r\nThe input will be a valid transition matrix for a game with n \u003e m ≥ 2. The output should be a double-precision scalar that has the average number of moves, accurate to at least 0.001.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 2373.27px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 644.075px 1186.64px; transform-origin: 644.081px 1186.64px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eBored of tedious court assemblies, King \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eNeduchadneddar\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e the Procrastinator has found a cunning way to keep his advisors, councilors, viziers, and other courtly lackeys busy: he plays a simple and pointless board game with them.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 42px; text-align: left; transform-origin: 385px 42px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ePlayers move their pawn along a path of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e squares, starting at square 1, with the goal of being the first to reach square \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. On each turn, they roll an \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e-sided die to determine how many squares to move. However, if their move would take them beyond the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eth\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e square, they stay put. Also, players cannot land on any of the numbers that King \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eNeduchadneddar\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e has decided are the unlucky numbers of the day. If their move would land them on any of these squares, they stay put instead.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eTo keep his courtiers on their toes (so they have less time to plot his downfall), the king changes the number of squares (\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e), the number of sides of the die (\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e), and the list of unlucky forbidden squares each day.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 31.5px; text-align: left; transform-origin: 385px 31.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eNeeding to know how much time he will waste with each game, the king in his wisdom (and/or insanity) has appointed you Chief Royal Mage of Matrix Computations and tasked you with determining the average number of moves it will take a player to complete a game according to today’s setup.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 73.6px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 36.8px; text-align: left; transform-origin: 385px 36.8px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eYou can use the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003etransition matrix\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e for the game to do this. The transition matrix \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eT\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is an \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e-by-\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e matrix where T(\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ej\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e,\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ek\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e) is the probability of moving from square \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ej\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e to square \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ek\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e in one move. Then \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"51\" height=\"21\" style=\"vertical-align: baseline;width: 51px;height: 21px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e gives the probability of moving from from square \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ej\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e to square \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ek\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e in \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e moves (where \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"20\" height=\"20\" style=\"vertical-align: baseline;width: 20px;height: 20px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is a matrix power; that is, \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"84\" height=\"18\" style=\"vertical-align: baseline;width: 84px;height: 18px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, using standard matrix multiplication).\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 95.6px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 47.8px; text-align: left; transform-origin: 385px 47.8px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cimg class=\"imageNode\" width=\"52\" height=\"21\" style=\"vertical-align: baseline;width: 52px;height: 21px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the probability of moving from square 1 to square \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e in \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e moves. That is, it is the probability of completing the game in \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e moves. However, once you reach square \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e you stay there forever. (In practice, the game would end at this point.) Therefore, \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"52\" height=\"21\" style=\"vertical-align: baseline;width: 52px;height: 21px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e increases monotonically towards 1 as \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e increases, and can be interpreted as the cumulative probability of completing the game in \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e moves. Or, in other words, the probability of completing the game in \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e moves or fewer.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 31.5px; text-align: left; transform-origin: 385px 31.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThis is convenient because there's a result in probability theory that allows you to calculate the expected value of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e (that is, the average number of moves to complete the game) from the cumulative probability. Skipping the math details, the punchline is:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 54.8px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 27.4px; text-align: left; transform-origin: 385px 27.4px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cimg class=\"imageNode\" width=\"136\" height=\"49\" style=\"vertical-align: baseline;width: 136px;height: 49px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 74.6px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 37.3px; text-align: left; transform-origin: 385px 37.3px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eAn infinite sum may not seem helpful, but remember that \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"52\" height=\"21\" style=\"vertical-align: baseline;width: 52px;height: 21px\" src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGgAAAAqCAYAAABfjB7GAAAAAXNSR0IArs4c6QAACU9JREFUeF7tWn2MXFUVP+e+menOLpTWaCvrdN57M5NuWJSoMfiBxhqpKW3jAtqaNeAHtTXEQgsSikLVghUhpECojbgBFqi1RRopETEh4CoRqR+RSNymZfbdO+22WhWCFHednTf3OGfznnmdnZl9s195JfP+adPee+6553fPx++cQWh9kbYARlq7lnLQAijij6AFUAugiFsg4uq1PKgFUMQtEHH1Wh7UAijiFoi4ei0PagEUcQtEXL1Z9aDOzs72efPmXQ4AywHgQgC4a2RkZG9HR8c1RHQjEZ2IxWI9ADDsum4vIt5aWXMWEa1USr0YcdvNiXqzChDfgEFKJBJ9iHixEKJXa70SEZ/QWl+OiF8hojWIuEpr/SvDMM4loh8g4jrHcR6cEwtE/JCaAJmmaQshngKA86aof0FrfUmhUDiUSqXeFo/HnyQiRMTfAsDuYrGY90B7DwAcJKLHLct6tlAo7ACAzyHiasdx/lDv7FQqlUwkEpuIaEhK+dN66yzLsoQQWyq6HFRK9U/xLjO5zbBt+2oAeMM0zT0DAwPuZMJrApTJZFYR0c8B4PmKVbfF4/HfHz58+NSyZctinhGvYcFEdLtS6mb+ayaTOYeIrgWArQAwQERrlVKvm6b5PiHELwFgIXuLUurJVCrVGY/HWf75AHC1lPKhVCq1wAOy6LrumuHh4ddqKb9kyZJOwzDuQ8QDUspH+ezqdQwMIt4AAOsAoC1KHsk2PHr06AYi6jIMY2s+n3+jEUi1AELLsrYj4ke01r2FQuFvvgDLshYg4mNeTgHvpbOnjX/e/z9KRC8HgLuKiB4AgL5isbj5xIkTI5ZlfRwRn0HE/W1tbesHBwff9IFExH7Hcb4BALpacdu2uwDgIUTscxyHPeI0cNLp9ELDMDYioqG1TiPil1lGlABifTyQthKRlUwmv8b3rwfSBIACRt5enagD3rCoEooOaa1XFQoF6Qv3840QYo/jOE/lcrl5ruvuQsQ1ALBaSvkbXmvb9jcrf34HES/jdfxvmUxmHEgiulQpdaBaYdM0zxVC/ISIXrAs61u1wgNffGBggIHVuVzu/HK5/DQALIkaQHy37u7us0ZHRznfqnQ6fVu9cDcBoGw2u4SIrhBC7Mjn88WgoXwjeuFtz9jY2Hr2iIAHtQkhtpXL5R8ycIFcNlTJF1dyyOvq6jp7bGyMQ9M7PTD+3t3dnRgdHd0JAOxZq7XWna7rvuyHOQZaa30vEV0ohOgZGho6NlnsNk3zPCEEA2RGESDW3zTN9wshngCAW6SUj9S6U+gqLmDE9R5Am5VS9zYylJ/LEPF6x3Hu4ZBk2/YFAPA0Iu72Q1k2m12ktWZPKhARh75RKeVuP8x5cn6GiJsdx9k1GTje5SMPkJfT7wCAT9R7eKEByuVy7yiXyxx6PgwAp4houVLqYANj+blsXbAqqxXKbNs2iWg/Ii5CxJvS6fRjvstzKBgZGXkYET+qtV5RKBT+/FYByAvtfkF2rZSSo8hpeTU0QJZlfZATOwCcDQAvlUql1cPDw8fDGGs6a7LZLBcrXAW+6FeGYeTNcojDVCq1MJFIdDMB11pfxIVJqVS6isNyOp3OGIZxEwB8FgDeRMTrHMfZX0vvQBp43TCMnnw+/8/gumYA2oSIHKb460smkxsHBwfHwhhrOmu8gmJ7s2fOJkC5XG6+1no5EfUCwGe8+3Eeud227VUAwGHrTwDAf18IAM/Ue1yByniZF5V+3TRAlmUxl7gfAL7Am+cq6S5evLijvb2dS3Qmr3c4jsOvMtQ3mwAFiiIuagYAoMTGFULEiOgGIcSG0dHRVz0y/nkAOJBIJK5kLlmtfKDTwutullJ+r2mAUqnUuzxi+V4A+EczuSCUNessWrp06dtLpRITWg6vTbV/5gggP6r8DhGZ12zQWm9m7uh3UCrec5FXpbHhJ5Bqfu+2bd9ZCYVMrB8hoq8qpf7rmyRUiPOJJQDEG7nrdMCotTeTySzVWv8CEbNRAyj48hFxLxEtqLSztvncMcDDmE5wQXVa6AreN5PJfJ+IttSybSiAAnngtPbOTANSLW86XjCdvWHuVdWv5NB1nZSSG7zjXmLbNhcI3Cf8o+u6PceOHTtRT24AoIPxeHz1kSNH/hXag4J5wMs/3Mj8f3snzGWmumY6Rp7O3jD6BvqVnJP3+i0r3lvVs7zPNM3rGzVGpwVQ1UuZ0N4Jc5mprgkQ2A9ELMT5HI97hqeEECuGhoZe8O9ZxRnXSCkfb2SDAEATiolJQ1zwpVTGD/tGRkbWnTx58j9TNXoz+wJtIR7qfVdKyZ3yUN9selCwaUxEtVpePmd8zR+71FO6ytsm0JfJAAq+FD5jQhkYylpTXxQ8/y4p5Y11KqEJJzQJEJ9zDgtRSv17sjOCTWPmQkqpvUEFLMvyq7txj3BdN1kh2x+SUnJFelqXPlhsENGE9llDgGqMFy52HOfZqdu7+Z0BD67LJWpJbQIgzGQyXyIi5nlcpT6QTCY3NxoBBJrGRwzDWJnP54d8Hap5jWEYO8vl8m2GYfwon8//tVrXQDl+QXWoHM/5DUzGr+rTiMiVCCvOyXDOAeIBXSwW4x5gR/V4oxHclmX18GjdW8MsvyYPqTIoLx8nnfXK4mDTuFZ4q+Ju/Vrr+UKIXfUetl+OE9ErtQaVEwBihdva2r5IRNym+JQPjnfRPA/ZAOCIEOLH1eOI5v0j1A5+4Vt4egsADROuN39aAQBrEZF/rNLmncDEj0ngc7FY7LmqfhcTxSsqLRnuzHNbpmGnJEjaaxUuXk5hTvNtIvoLd/KllM/XC5u+N9YrgibLQaEsONuLeEaltT5ARIeqZ1AzeTZTimQy2QcA9zciljN1Jv9MoDKI7iei9lKp1FtrzH9GAMQGsW2bp7L9iLh2tnhYNpt9NxF9HQB47sTFwqx+3p12ElFPvZ+ZnTEABeb4n3Rdd20jZj4Vq3oj9XsMw7i1VjKfisxGezj3uK67DxHvrPfjl8mKhJnWadryOMeUy+VbKtPYXLFY3Hj8+PFXpy0UAHh+I4TgLvTdjuO8MhMyG8nwHsODRPSwUmpfo7L+jPEg/8LsSUopntJ+bGxsbNNMgTTboPjyuTNjGMYOIrq7UfHgrz/jAPIV9179fKXUS3Nl3Bk4B70JcV5KeTKMvDMWoDCXeyusaQEUcRRbAEUcoP8Bgqk+dnshbtUAAAAASUVORK5CYII=\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e increases monotonically towards 1, which means the terms in the sum go to zero. Therefore, in practice, you can find the average number of moves to complete the game by summing \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"76\" height=\"21\" style=\"vertical-align: baseline;width: 76px;height: 21px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e until the terms are too small to make any practical difference.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 24.8px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 12.4px; text-align: left; transform-origin: 385px 12.4px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eGiven a transition matrix \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eT\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, return the average number of moves \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"14\" height=\"19\" style=\"vertical-align: baseline;width: 14px;height: 19px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, accurate to 3 decimal places.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExample\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWith \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e = 8 and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e = 3, and square 5 forbidden, the transition matrix is:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 245.25px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 641.075px 122.625px; transform-origin: 641.081px 122.625px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eT = [0      1/3    1/3    1/3    0      0      0      0       \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    0      1/3    1/3    1/3    0      0      0      0       \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    0      0      1/3    1/3    0      1/3    0      0       \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    0      0      0      1/3    0      1/3    1/3    0       \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    0      0      0      0      0      0      0      0       \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    0      0      0      0      0      1/3    1/3    1/3     \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    0      0      0      0      0      0      2/3    1/3     \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    0      0      0      0      0      0      0      1];\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eavgm = avgmoves(T)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eavgm =\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e6.3750\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eAn inexact way to verify the result is to directly simulate playing the game many times:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 551.812px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 641.075px 275.9px; transform-origin: 641.081px 275.906px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"border-block-end-color: rgb(14, 0, 255); border-block-start-color: rgb(14, 0, 255); border-bottom-color: rgb(14, 0, 255); border-inline-end-color: rgb(14, 0, 255); border-inline-start-color: rgb(14, 0, 255); border-left-color: rgb(14, 0, 255); border-right-color: rgb(14, 0, 255); border-top-color: rgb(14, 0, 255); caret-color: rgb(14, 0, 255); color: rgb(14, 0, 255); column-rule-color: rgb(14, 0, 255); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(14, 0, 255); text-decoration-color: rgb(14, 0, 255); text-emphasis-color: rgb(14, 0, 255); \"\u003efunction \u003c/span\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eavgm = simulategame(n,m,nogo,nexp)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003emv = zeros(nexp,1);\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"border-block-end-color: rgb(14, 0, 255); border-block-start-color: rgb(14, 0, 255); border-bottom-color: rgb(14, 0, 255); border-inline-end-color: rgb(14, 0, 255); border-inline-start-color: rgb(14, 0, 255); border-left-color: rgb(14, 0, 255); border-right-color: rgb(14, 0, 255); border-top-color: rgb(14, 0, 255); caret-color: rgb(14, 0, 255); color: rgb(14, 0, 255); column-rule-color: rgb(14, 0, 255); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(14, 0, 255); text-decoration-color: rgb(14, 0, 255); text-emphasis-color: rgb(14, 0, 255); \"\u003efor \u003c/span\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eg = 1:nexp\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    \u003c/span\u003e\u003cspan style=\"border-block-end-color: rgb(2, 128, 9); border-block-start-color: rgb(2, 128, 9); border-bottom-color: rgb(2, 128, 9); border-inline-end-color: rgb(2, 128, 9); border-inline-start-color: rgb(2, 128, 9); border-left-color: rgb(2, 128, 9); border-right-color: rgb(2, 128, 9); border-top-color: rgb(2, 128, 9); caret-color: rgb(2, 128, 9); color: rgb(2, 128, 9); column-rule-color: rgb(2, 128, 9); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(2, 128, 9); text-decoration-color: rgb(2, 128, 9); text-emphasis-color: rgb(2, 128, 9); \"\u003e% Start game at 1\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    s = 1;\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    \u003c/span\u003e\u003cspan style=\"border-block-end-color: rgb(2, 128, 9); border-block-start-color: rgb(2, 128, 9); border-bottom-color: rgb(2, 128, 9); border-inline-end-color: rgb(2, 128, 9); border-inline-start-color: rgb(2, 128, 9); border-left-color: rgb(2, 128, 9); border-right-color: rgb(2, 128, 9); border-top-color: rgb(2, 128, 9); caret-color: rgb(2, 128, 9); color: rgb(2, 128, 9); column-rule-color: rgb(2, 128, 9); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(2, 128, 9); text-decoration-color: rgb(2, 128, 9); text-emphasis-color: rgb(2, 128, 9); \"\u003e% Go until we reach the nth square\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    \u003c/span\u003e\u003cspan style=\"border-block-end-color: rgb(14, 0, 255); border-block-start-color: rgb(14, 0, 255); border-bottom-color: rgb(14, 0, 255); border-inline-end-color: rgb(14, 0, 255); border-inline-start-color: rgb(14, 0, 255); border-left-color: rgb(14, 0, 255); border-right-color: rgb(14, 0, 255); border-top-color: rgb(14, 0, 255); caret-color: rgb(14, 0, 255); color: rgb(14, 0, 255); column-rule-color: rgb(14, 0, 255); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(14, 0, 255); text-decoration-color: rgb(14, 0, 255); text-emphasis-color: rgb(14, 0, 255); \"\u003ewhile \u003c/span\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e(s \u0026lt; n)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e        \u003c/span\u003e\u003cspan style=\"border-block-end-color: rgb(2, 128, 9); border-block-start-color: rgb(2, 128, 9); border-bottom-color: rgb(2, 128, 9); border-inline-end-color: rgb(2, 128, 9); border-inline-start-color: rgb(2, 128, 9); border-left-color: rgb(2, 128, 9); border-right-color: rgb(2, 128, 9); border-top-color: rgb(2, 128, 9); caret-color: rgb(2, 128, 9); color: rgb(2, 128, 9); column-rule-color: rgb(2, 128, 9); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(2, 128, 9); text-decoration-color: rgb(2, 128, 9); text-emphasis-color: rgb(2, 128, 9); \"\u003e% Roll die and see where we would land\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e        snext = s + randi(m);\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e        \u003c/span\u003e\u003cspan style=\"border-block-end-color: rgb(14, 0, 255); border-block-start-color: rgb(14, 0, 255); border-bottom-color: rgb(14, 0, 255); border-inline-end-color: rgb(14, 0, 255); border-inline-start-color: rgb(14, 0, 255); border-left-color: rgb(14, 0, 255); border-right-color: rgb(14, 0, 255); border-top-color: rgb(14, 0, 255); caret-color: rgb(14, 0, 255); color: rgb(14, 0, 255); column-rule-color: rgb(14, 0, 255); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(14, 0, 255); text-decoration-color: rgb(14, 0, 255); text-emphasis-color: rgb(14, 0, 255); \"\u003eif \u003c/span\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e(snext \u0026lt;= n) \u0026amp;\u0026amp; ~any(snext == nogo)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e            \u003c/span\u003e\u003cspan style=\"border-block-end-color: rgb(2, 128, 9); border-block-start-color: rgb(2, 128, 9); border-bottom-color: rgb(2, 128, 9); border-inline-end-color: rgb(2, 128, 9); border-inline-start-color: rgb(2, 128, 9); border-left-color: rgb(2, 128, 9); border-right-color: rgb(2, 128, 9); border-top-color: rgb(2, 128, 9); caret-color: rgb(2, 128, 9); color: rgb(2, 128, 9); column-rule-color: rgb(2, 128, 9); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(2, 128, 9); text-decoration-color: rgb(2, 128, 9); text-emphasis-color: rgb(2, 128, 9); \"\u003e% Move only if valid\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e            s = snext;\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e        \u003c/span\u003e\u003cspan style=\"border-block-end-color: rgb(14, 0, 255); border-block-start-color: rgb(14, 0, 255); border-bottom-color: rgb(14, 0, 255); border-inline-end-color: rgb(14, 0, 255); border-inline-start-color: rgb(14, 0, 255); border-left-color: rgb(14, 0, 255); border-right-color: rgb(14, 0, 255); border-top-color: rgb(14, 0, 255); caret-color: rgb(14, 0, 255); color: rgb(14, 0, 255); column-rule-color: rgb(14, 0, 255); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(14, 0, 255); text-decoration-color: rgb(14, 0, 255); text-emphasis-color: rgb(14, 0, 255); \"\u003eend\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e        \u003c/span\u003e\u003cspan style=\"border-block-end-color: rgb(2, 128, 9); border-block-start-color: rgb(2, 128, 9); border-bottom-color: rgb(2, 128, 9); border-inline-end-color: rgb(2, 128, 9); border-inline-start-color: rgb(2, 128, 9); border-left-color: rgb(2, 128, 9); border-right-color: rgb(2, 128, 9); border-top-color: rgb(2, 128, 9); caret-color: rgb(2, 128, 9); color: rgb(2, 128, 9); column-rule-color: rgb(2, 128, 9); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(2, 128, 9); text-decoration-color: rgb(2, 128, 9); text-emphasis-color: rgb(2, 128, 9); \"\u003e% Increment number of moves for this game\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e        mv(g) = mv(g) + 1;\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    \u003c/span\u003e\u003cspan style=\"border-block-end-color: rgb(14, 0, 255); border-block-start-color: rgb(14, 0, 255); border-bottom-color: rgb(14, 0, 255); border-inline-end-color: rgb(14, 0, 255); border-inline-start-color: rgb(14, 0, 255); border-left-color: rgb(14, 0, 255); border-right-color: rgb(14, 0, 255); border-top-color: rgb(14, 0, 255); caret-color: rgb(14, 0, 255); color: rgb(14, 0, 255); column-rule-color: rgb(14, 0, 255); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(14, 0, 255); text-decoration-color: rgb(14, 0, 255); text-emphasis-color: rgb(14, 0, 255); \"\u003eend\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    \u003c/span\u003e\u003cspan style=\"border-block-end-color: rgb(2, 128, 9); border-block-start-color: rgb(2, 128, 9); border-bottom-color: rgb(2, 128, 9); border-inline-end-color: rgb(2, 128, 9); border-inline-start-color: rgb(2, 128, 9); border-left-color: rgb(2, 128, 9); border-right-color: rgb(2, 128, 9); border-top-color: rgb(2, 128, 9); caret-color: rgb(2, 128, 9); color: rgb(2, 128, 9); column-rule-color: rgb(2, 128, 9); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(2, 128, 9); text-decoration-color: rgb(2, 128, 9); text-emphasis-color: rgb(2, 128, 9); \"\u003e% Go to the next game\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"border-block-end-color: rgb(14, 0, 255); border-block-start-color: rgb(14, 0, 255); border-bottom-color: rgb(14, 0, 255); border-inline-end-color: rgb(14, 0, 255); border-inline-start-color: rgb(14, 0, 255); border-left-color: rgb(14, 0, 255); border-right-color: rgb(14, 0, 255); border-top-color: rgb(14, 0, 255); caret-color: rgb(14, 0, 255); color: rgb(14, 0, 255); column-rule-color: rgb(14, 0, 255); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(14, 0, 255); text-decoration-color: rgb(14, 0, 255); text-emphasis-color: rgb(14, 0, 255); \"\u003eend\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eavgm = mean(mv);\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"border-block-end-color: rgb(14, 0, 255); border-block-start-color: rgb(14, 0, 255); border-bottom-color: rgb(14, 0, 255); border-inline-end-color: rgb(14, 0, 255); border-inline-start-color: rgb(14, 0, 255); border-left-color: rgb(14, 0, 255); border-right-color: rgb(14, 0, 255); border-top-color: rgb(14, 0, 255); caret-color: rgb(14, 0, 255); color: rgb(14, 0, 255); column-rule-color: rgb(14, 0, 255); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(14, 0, 255); text-decoration-color: rgb(14, 0, 255); text-emphasis-color: rgb(14, 0, 255); \"\u003eend\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003en = 8;\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003em = 3;\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003enogo = 5;\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eavgmsim = simulategame(n,m,nogo,1e4)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eavgmsim =\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e6.3626\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThis approach would require too many simulations to be accurate to the required precision, but gives rough verification that the game takes 6.375 moves on average.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor n = 16, m = 5, and the forbidden squares are 4, 6, 10, 11, and 15, the transition matrix is:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 429.188px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 641.075px 214.587px; transform-origin: 641.081px 214.594px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eidx = [1;17;18;33;34;35;65;66;67;69;98;99;101;103;115;117;119;120;133;135;136;137;183;184;185;188;200;201;204;205;217;220;221;222;252;253;254;256];\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eTvals = [0.4;0.2;0.4;0.2;0.2;0.4;0.2;0.2;0.2;0.4;0.2;0.2;0.2;0.4;0.2;0.2;0.2;0.4;0.2;0.2;0.2;0.4;0.2;0.2;0.2;0.4;0.2;0.2;0.2;0.6;0.2;0.2;0.2;0.8;0.2;0.2;0.2;1];\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eT = zeros(16);\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eT(idx) = Tvals\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eT =\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e0.4000    0.2000    0.2000         0    0.2000         0         0         0         0         0         0         0         0         0         0         0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e0    0.4000    0.2000         0    0.2000         0    0.2000         0         0         0         0         0         0         0         0         0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e0         0    0.4000         0    0.2000         0    0.2000    0.2000         0         0         0         0         0         0         0         0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e0         0         0         0    0.4000         0    0.2000    0.2000    0.2000         0         0         0         0         0         0         0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e0         0         0         0         0         0    0.4000    0.2000    0.2000         0         0    0.2000         0         0         0         0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e0         0         0         0         0         0         0    0.4000    0.2000         0         0    0.2000    0.2000         0         0         0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e0         0         0         0         0         0         0         0    0.4000         0         0    0.2000    0.2000    0.2000         0         0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e0         0         0         0         0         0         0         0         0         0         0    0.4000    0.2000    0.2000         0    0.2000\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e0         0         0         0         0         0         0         0         0         0         0         0    0.6000    0.2000         0    0.2000\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e0         0         0         0         0         0         0         0         0         0         0         0         0    0.8000         0    0.2000\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e0         0         0         0         0         0         0         0         0         0         0         0         0         0         0    1.0000\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eComputing the average number of moves from the transition matrix and from simulation:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 122.625px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 641.075px 61.3125px; transform-origin: 641.081px 61.3125px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eavgm = avgmoves(T)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eavgm =\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e11.4015\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eavgmsim = simulategame(16,5,[4 6 10 11 15],1e4)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eavgmsim =\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e11.4266\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eAssumptions\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe input will be a valid transition matrix for a game with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u0026gt; \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e ≥ 2. The output should be a double-precision scalar that has the average number of moves, accurate to at least 0.001.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function mv = avgmoves(T)\r\nmv = 1 + (1-T(1,end)); % m=0 + m=1\r\nend","test_suite":"%% Test case #1: m = 3, n = 28, nogo = [3 7 13 18 21]\r\nidx = [1;29;30;85;86;88;114;116;117;144;145;146;201;202;230;232;260;261;262;288;289;290;291;317;318;319;320;375;376;404;406;407;434;435;436;462;463;464;465;520;521;523;549;551;552;607;608;636;638;666;667;694;695;696;723;724;725;726;752;753;754;755;781;782;783;784];\r\nTvals = [1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;2;1;1;1;3]/3;\r\nT = zeros(28);\r\nT(idx) = Tvals;\r\nmavg = avgmoves(T);\r\nassert( abs(mavg - 19.09765625) \u003c 0.001 )\r\n\r\n%% Test case #2: m = 3, n = 25, nogo = [8 17 18 22]\r\nidx = [26;51;52;76;77;78;102;103;104;105;128;129;130;131;154;155;156;157;206;207;232;234;259;260;284;285;286;310;311;312;336;337;338;339;362;363;364;365;388;389;390;391;466;469;494;495;519;520;521;570;571;573;596;598;599;623;624;625];\r\nTvals = [1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;2;1;1;1;2;1;1;1;1;1;1;1;1;1;1;1;1;2;1;1;3]/3;\r\nT = zeros(25);\r\nT(idx) = Tvals;\r\nmavg = avgmoves(T);\r\nassert( abs(mavg - 19.52023320) \u003c 0.001 )\r\n\r\n%% Test case #3: m = 4, n = 10, nogo = [4 5 6]\r\nidx = [1;11;12;21;22;23;63;67;77;78;87;88;89;97;98;99;100];\r\nTvals = [2;1;3;1;1;3;1;1;1;2;1;1;3;1;1;1;4]/4;\r\nT = zeros(10);\r\nT(idx) = Tvals;\r\nmavg = avgmoves(T);\r\nassert( abs(mavg - 12.00000000) \u003c 0.001 )\r\n\r\n%% Test case #4: m = 4, n = 26, nogo = [3 13]\r\nidx = [1;27;28;79;80;105;106;108;132;134;135;160;161;162;186;187;188;189;213;214;215;216;217;240;241;242;243;244;267;268;269;270;271;294;295;296;297;298;348;349;350;375;376;378;402;404;405;430;431;432;456;457;458;459;483;484;485;486;510;511;512;513;537;538;539;540;564;565;566;567;591;592;593;594;595;618;619;620;621;622;645;646;647;648;649;672;673;674;675;676];\r\nTvals = [1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;2;1;1;1;1;3;1;1;1;1;4]/4;\r\nT = zeros(26);\r\nT(idx) = Tvals;\r\nmavg = avgmoves(T);\r\nassert( abs(mavg - 13.69784774) \u003c 0.001 )\r\n\r\n%% Test case #5: m = 5, n = 12, nogo = [5 6 8 9 11]\r\nidx = [1;13;14;25;26;27;37;38;39;40;74;75;76;79;115;118;139;142;144];\r\nTvals = [2;1;2;1;1;3;1;1;1;4;1;1;1;3;1;4;1;1;5]/5;\r\nT = zeros(12);\r\nT(idx) = Tvals;\r\nmavg = avgmoves(T);\r\nassert( abs(mavg - 11.66666667) \u003c 0.001 )\r\n\r\n%% Test case #6: m = 5, n = 18, nogo = [5 6 9 10]\r\nidx = [1;19;20;37;38;39;55;56;57;58;110;111;112;115;129;130;133;134;187;188;205;206;209;224;227;228;245;246;247;248;263;264;265;266;267;281;282;283;284;285;286;300;301;302;303;304;305;319;320;321;322;323;324];\r\nTvals = [2;1;2;1;1;2;1;1;1;3;1;1;1;2;1;1;1;2;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;2;1;1;1;1;1;3;1;1;1;1;1;4;1;1;1;1;1;5]/5;\r\nT = zeros(18);\r\nT(idx) = Tvals;\r\nmavg = avgmoves(T);\r\nassert( abs(mavg - 12.28888889) \u003c 0.001 )\r\n\r\n%% Test case #7: m = 6, n = 8, nogo = []\r\nidx = [9;17;18;19;25;26;27;28;33;34;35;36;37;41;42;43;44;45;46;49;50;51;52;53;54;55;58;59;60;61;62;63;64];\r\nTvals = [1;1;1;1;1;1;1;2;1;1;1;1;3;1;1;1;1;1;4;1;1;1;1;1;1;5;1;1;1;1;1;1;6]/6;\r\nT = zeros(8);\r\nT(idx) = Tvals;\r\nmavg = avgmoves(T);\r\nassert( abs(mavg - 7.00000000) \u003c 0.001 )\r\n\r\n%% Test case #8: m = 6, n = 18, nogo = []\r\nidx = [19;37;38;55;56;57;73;74;75;76;91;92;93;94;95;109;110;111;112;113;114;128;129;130;131;132;133;147;148;149;150;151;152;166;167;168;169;170;171;185;186;187;188;189;190;204;205;206;207;208;209;223;224;225;226;227;228;229;242;243;244;245;246;247;248;261;262;263;264;265;266;267;280;281;282;283;284;285;286;299;300;301;302;303;304;305;318;319;320;321;322;323;324];\r\nTvals = [1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;2;1;1;1;1;1;1;3;1;1;1;1;1;1;4;1;1;1;1;1;1;5;1;1;1;1;1;1;6]/6;\r\nT = zeros(18);\r\nT(idx) = Tvals;\r\nmavg = avgmoves(T);\r\nassert( abs(mavg - 9.61298219) \u003c 0.001 )\r\n\r\n%% Test case #9: m = 7, n = 21, nogo = 3\r\nidx = [1;22;23;64;65;85;86;88;106;107;109;110;127;128;130;131;132;148;149;151;152;153;154;170;172;173;174;175;176;193;194;195;196;197;198;214;215;216;217;218;219;220;236;237;238;239;240;241;242;258;259;260;261;262;263;264;280;281;282;283;284;285;286;302;303;304;305;306;307;308;309;324;325;326;327;328;329;330;331;346;347;348;349;350;351;352;353;368;369;370;371;372;373;374;375;390;391;392;393;394;395;396;397;412;413;414;415;416;417;418;419;434;435;436;437;438;439;440;441];\r\nTvals = [1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;2;1;1;1;1;1;1;1;3;1;1;1;1;1;1;1;4;1;1;1;1;1;1;1;5;1;1;1;1;1;1;1;6;1;1;1;1;1;1;1;7]/7;\r\nT = zeros(21);\r\nT(idx) = Tvals;\r\nmavg = avgmoves(T);\r\nassert( abs(mavg - 10.83736048) \u003c 0.001 )\r\n\r\n%% Test case #10: m = 7, n = 9, nogo = [2 8]\r\nidx = [1;19;21;28;30;31;37;39;40;41;46;48;49;50;51;55;57;58;59;60;61;75;76;77;78;79;81];\r\nTvals = [2;1;2;1;1;3;1;1;1;4;1;1;1;1;5;1;1;1;1;1;6;1;1;1;1;1;7]/7;\r\nT = zeros(9);\r\nT(idx) = Tvals;\r\nmavg = avgmoves(T);\r\nassert( abs(mavg - 8.40000000) \u003c 0.001 )\r\n\r\n%% Test case #11: m = 8, n = 17, nogo = [4 6 9 16]\r\nidx = [1;18;19;35;36;37;69;70;71;73;103;104;105;107;109;120;121;122;124;126;127;155;156;158;160;161;163;173;175;177;178;180;181;192;194;195;197;198;199;209;211;212;214;215;216;217;228;229;231;232;233;234;235;245;246;248;249;250;251;252;253;282;283;284;285;286;287;289];\r\nTvals = [3;1;3;1;1;3;1;1;1;2;1;1;1;1;1;1;1;1;1;1;2;1;1;1;1;1;2;1;1;1;1;1;3;1;1;1;1;1;4;1;1;1;1;1;1;5;1;1;1;1;1;1;6;1;1;1;1;1;1;1;7;1;1;1;1;1;1;8]/8;\r\nT = zeros(17);\r\nT(idx) = Tvals;\r\nmavg = avgmoves(T);\r\nassert( abs(mavg - 11.58400000) \u003c 0.001 )\r\n\r\n%% Test case #12: m = 8, n = 9, nogo = [3 5 7]\r\nidx = [1;10;11;28;29;31;46;47;49;51;64;65;67;69;71;73;74;76;78;80;81];\r\nTvals = [3;1;4;1;1;5;1;1;1;6;1;1;1;1;7;1;1;1;1;1;8]/8;\r\nT = zeros(9);\r\nT(idx) = Tvals;\r\nmavg = avgmoves(T);\r\nassert( abs(mavg - 8.00000000) \u003c 0.001 )\r\n\r\n%% Test case #13: m = 9, n = 19, nogo = [8 9 13]\r\nidx = [1;20;21;39;40;41;58;59;60;61;77;78;79;80;81;96;97;98;99;100;101;115;116;117;118;119;120;121;172;173;174;175;176;177;178;181;192;193;194;195;196;197;200;201;212;213;214;215;216;219;220;221;252;253;254;257;258;259;261;272;273;276;277;278;280;281;292;295;296;297;299;300;301;314;315;316;318;319;320;321;333;334;335;337;338;339;340;341;352;353;354;356;357;358;359;360;361];\r\nTvals = [2;1;2;1;1;2;1;1;1;3;1;1;1;1;3;1;1;1;1;1;3;1;1;1;1;1;1;3;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;2;1;1;1;1;1;1;1;3;1;1;1;1;1;1;4;1;1;1;1;1;1;5;1;1;1;1;1;1;6;1;1;1;1;1;1;7;1;1;1;1;1;1;1;8;1;1;1;1;1;1;1;1;9]/9;\r\nT = zeros(19);\r\nT(idx) = Tvals;\r\nmavg = avgmoves(T);\r\nassert( abs(mavg - 12.11111111) \u003c 0.001 )\r\n\r\n%% Test case #14: m = 9, n = 29, nogo = 13\r\nidx = [30;59;60;88;89;90;91;117;118;119;120;121;146;147;148;149;150;151;175;176;177;178;179;180;181;204;205;206;207;208;209;210;211;233;234;235;236;237;238;239;240;241;262;263;264;265;266;267;268;269;270;271;292;293;294;295;296;297;298;299;300;301;322;323;324;325;326;327;328;329;330;331;382;383;384;385;386;387;388;389;412;413;414;415;416;417;418;420;442;443;444;445;446;447;449;450;472;473;474;475;476;478;479;480;502;503;504;505;507;508;509;510;532;533;534;536;537;538;539;540;562;563;565;566;567;568;569;570;592;594;595;596;597;598;599;600;601;623;624;625;626;627;628;629;630;631;652;653;654;655;656;657;658;659;660;661;682;683;684;685;686;687;688;689;690;691;712;713;714;715;716;717;718;719;720;721;742;743;744;745;746;747;748;749;750;751;772;773;774;775;776;777;778;779;780;781;802;803;804;805;806;807;808;809;810;811;832;833;834;835;836;837;838;839;840;841];\r\nTvals = [1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;2;1;1;1;1;1;1;1;1;1;3;1;1;1;1;1;1;1;1;1;4;1;1;1;1;1;1;1;1;1;5;1;1;1;1;1;1;1;1;1;6;1;1;1;1;1;1;1;1;1;7;1;1;1;1;1;1;1;1;1;8;1;1;1;1;1;1;1;1;1;9]/9;\r\nT = zeros(29);\r\nT(idx) = Tvals;\r\nmavg = avgmoves(T);\r\nassert( abs(mavg - 13.56167777) \u003c 0.001 )\r\n\r\n%% Test case #15: m = 10, n = 15, nogo = [7 10]\r\nidx = [1;16;17;31;32;33;46;47;48;49;61;62;63;64;65;76;77;78;79;80;81;106;107;108;109;110;111;113;121;122;123;124;125;126;128;129;151;152;153;154;155;156;158;159;161;167;168;169;170;171;173;174;176;177;183;184;185;186;188;189;191;192;193;199;200;201;203;204;206;207;208;209;215;216;218;219;221;222;223;224;225];\r\nTvals = [2;1;2;1;1;2;1;1;1;2;1;1;1;1;2;1;1;1;1;1;3;1;1;1;1;1;1;4;1;1;1;1;1;1;1;5;1;1;1;1;1;1;1;1;6;1;1;1;1;1;1;1;1;7;1;1;1;1;1;1;1;1;8;1;1;1;1;1;1;1;1;9;1;1;1;1;1;1;1;1;10]/10;\r\nT = zeros(15);\r\nT(idx) = Tvals;\r\nmavg = avgmoves(T);\r\nassert( abs(mavg - 11.77978516) \u003c 0.001 )\r\n\r\n%% Test case #16: m = 10, n = 23, nogo = 14\r\nidx = [24;47;48;70;71;72;73;93;94;95;96;97;116;117;118;119;120;121;139;140;141;142;143;144;145;162;163;164;165;166;167;168;169;185;186;187;188;189;190;191;192;193;208;209;210;211;212;213;214;215;216;217;231;232;233;234;235;236;237;238;239;240;241;255;256;257;258;259;260;261;262;263;264;265;279;280;281;282;283;284;285;286;287;288;289;327;328;329;330;331;332;333;334;335;337;351;352;353;354;355;356;357;358;360;361;375;376;377;378;379;380;381;383;384;385;399;400;401;402;403;404;406;407;408;409;423;424;425;426;427;429;430;431;432;433;447;448;449;450;452;453;454;455;456;457;471;472;473;475;476;477;478;479;480;481;495;496;498;499;500;501;502;503;504;505;519;521;522;523;524;525;526;527;528;529];\r\nTvals = [1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;2;1;1;1;1;1;1;1;1;1;3;1;1;1;1;1;1;1;1;1;4;1;1;1;1;1;1;1;1;1;5;1;1;1;1;1;1;1;1;1;6;1;1;1;1;1;1;1;1;1;7;1;1;1;1;1;1;1;1;1;8;1;1;1;1;1;1;1;1;1;9;1;1;1;1;1;1;1;1;1;10]/10;\r\nT = zeros(23);\r\nT(idx) = Tvals;\r\nmavg = avgmoves(T);\r\nassert( abs(mavg - 13.01211039) \u003c 0.001 )\r\n\r\n%% Test case #17: m = 11, n = 13, nogo = [6 7 11]\r\nidx = [1;14;15;27;28;29;40;41;42;43;53;54;55;56;57;92;93;94;95;96;99;105;106;107;108;109;112;113;118;119;120;121;122;125;126;127;144;145;146;147;148;151;152;153;155;158;159;160;161;164;165;166;168;169];\r\nTvals = [3;1;3;1;1;4;1;1;1;5;1;1;1;1;6;1;1;1;1;1;7;1;1;1;1;1;1;8;1;1;1;1;1;1;1;9;1;1;1;1;1;1;1;1;10;1;1;1;1;1;1;1;1;11]/11;\r\nT = zeros(13);\r\nT(idx) = Tvals;\r\nmavg = avgmoves(T);\r\nassert( abs(mavg - 12.37500000) \u003c 0.001 )\r\n\r\n%% Test case #18: m = 11, n = 18, nogo = []\r\nidx = [19;37;38;55;56;57;73;74;75;76;91;92;93;94;95;109;110;111;112;113;114;127;128;129;130;131;132;133;134;145;146;147;148;149;150;151;152;153;163;164;165;166;167;168;169;170;171;172;181;182;183;184;185;186;187;188;189;190;191;199;200;201;202;203;204;205;206;207;208;209;210;218;219;220;221;222;223;224;225;226;227;228;229;237;238;239;240;241;242;243;244;245;246;247;248;256;257;258;259;260;261;262;263;264;265;266;267;275;276;277;278;279;280;281;282;283;284;285;286;294;295;296;297;298;299;300;301;302;303;304;305;313;314;315;316;317;318;319;320;321;322;323;324];\r\nTvals = [1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;2;1;1;1;1;1;1;1;1;1;3;1;1;1;1;1;1;1;1;1;1;4;1;1;1;1;1;1;1;1;1;1;1;5;1;1;1;1;1;1;1;1;1;1;1;6;1;1;1;1;1;1;1;1;1;1;1;7;1;1;1;1;1;1;1;1;1;1;1;8;1;1;1;1;1;1;1;1;1;1;1;9;1;1;1;1;1;1;1;1;1;1;1;10;1;1;1;1;1;1;1;1;1;1;1;11]/11;\r\nT = zeros(18);\r\nT(idx) = Tvals;\r\nmavg = avgmoves(T);\r\nassert( abs(mavg - 12.54505095) \u003c 0.001 )\r\n\r\n%% Test case #19: m = 12, n = 19, nogo = [2 10 18]\r\nidx = [1;39;41;58;60;61;77;79;80;81;96;98;99;100;101;115;117;118;119;120;121;134;136;137;138;139;140;141;153;155;156;157;158;159;160;161;191;193;194;195;196;197;198;199;201;210;212;213;214;215;216;217;218;220;221;229;231;232;233;234;235;236;237;239;240;241;250;251;252;253;254;255;256;258;259;260;261;269;270;271;272;273;274;275;277;278;279;280;281;289;290;291;292;293;294;296;297;298;299;300;301;309;310;311;312;313;315;316;317;318;319;320;321;349;350;351;353;354;355;356;357;358;359;361];\r\nTvals = [2;1;1;1;1;1;1;1;1;1;1;1;1;1;2;1;1;1;1;1;2;1;1;1;1;1;1;3;1;1;1;1;1;1;1;4;1;1;1;1;1;1;1;1;5;1;1;1;1;1;1;1;1;1;6;1;1;1;1;1;1;1;1;1;1;7;1;1;1;1;1;1;1;1;1;1;8;1;1;1;1;1;1;1;1;1;1;1;9;1;1;1;1;1;1;1;1;1;1;1;10;1;1;1;1;1;1;1;1;1;1;1;11;1;1;1;1;1;1;1;1;1;1;12]/12;\r\nT = zeros(19);\r\nT(idx) = Tvals;\r\nmavg = avgmoves(T);\r\nassert( abs(mavg - 13.71371901) \u003c 0.001 )\r\n\r\n%% Test case #20: m = 12, n = 23, nogo = [6 7 11 14]\r\nidx = [1;24;25;47;48;49;70;71;72;73;93;94;95;96;97;162;163;164;165;166;169;185;186;187;188;189;192;193;208;209;210;211;212;215;216;217;254;255;256;257;258;261;262;263;265;277;278;279;280;281;284;285;286;288;289;325;326;327;330;331;332;334;335;337;349;350;353;354;355;357;358;360;361;373;376;377;378;380;381;383;384;385;399;400;401;403;404;406;407;408;409;422;423;424;426;427;429;430;431;432;433;445;446;447;449;450;452;453;454;455;456;457;469;470;472;473;475;476;477;478;479;480;481;493;495;496;498;499;500;501;502;503;504;505;518;519;521;522;523;524;525;526;527;528;529];\r\nTvals = [3;1;4;1;1;4;1;1;1;4;1;1;1;1;4;1;1;1;1;1;2;1;1;1;1;1;1;2;1;1;1;1;1;1;1;2;1;1;1;1;1;1;1;1;2;1;1;1;1;1;1;1;1;1;3;1;1;1;1;1;1;1;1;4;1;1;1;1;1;1;1;1;5;1;1;1;1;1;1;1;1;6;1;1;1;1;1;1;1;1;7;1;1;1;1;1;1;1;1;1;8;1;1;1;1;1;1;1;1;1;1;9;1;1;1;1;1;1;1;1;1;1;10;1;1;1;1;1;1;1;1;1;1;11;1;1;1;1;1;1;1;1;1;1;12]/12;\r\nT = zeros(23);\r\nT(idx) = Tvals;\r\nmavg = avgmoves(T);\r\nassert( abs(mavg - 14.84267285) \u003c 0.001 )","published":true,"deleted":false,"likes_count":3,"comments_count":0,"created_by":287,"edited_by":287,"edited_at":"2025-11-10T02:21:25.000Z","deleted_by":null,"deleted_at":null,"solvers_count":82,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2025-11-04T20:41:43.000Z","updated_at":"2026-02-13T10:03:15.000Z","published_at":"2025-11-10T02:21:25.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eBored of tedious court assemblies, King \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eNeduchadneddar\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e the Procrastinator has found a cunning way to keep his advisors, councilors, viziers, and other courtly lackeys busy: he plays a simple and pointless board game with them.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ePlayers move their pawn along a path of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e squares, starting at square 1, with the goal of being the first to reach square \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e. On each turn, they roll an \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e-sided die to determine how many squares to move. However, if their move would take them beyond the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eth\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e square, they stay put. Also, players cannot land on any of the numbers that King \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eNeduchadneddar\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e has decided are the unlucky numbers of the day. If their move would land them on any of these squares, they stay put instead.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eTo keep his courtiers on their toes (so they have less time to plot his downfall), the king changes the number of squares (\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e), the number of sides of the die (\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e), and the list of unlucky forbidden squares each day.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eNeeding to know how much time he will waste with each game, the king in his wisdom (and/or insanity) has appointed you Chief Royal Mage of Matrix Computations and tasked you with determining the average number of moves it will take a player to complete a game according to today’s setup.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eYou can use the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003etransition matrix\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e for the game to do this. The transition matrix \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eT\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e is an \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e-by-\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e matrix where T(\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ej\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e,\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ek\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e) is the probability of moving from square \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ej\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e to square \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ek\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e in one move. Then \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"21\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"51\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e gives the probability of moving from from square \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ej\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e to square \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ek\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e in \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e moves (where \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"20\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"20\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is a matrix power; that is, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"84\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId3\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, using standard matrix multiplication).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"21\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"52\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId4\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the probability of moving from square 1 to square \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e in \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e moves. That is, it is the probability of completing the game in \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e moves. However, once you reach square \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e you stay there forever. (In practice, the game would end at this point.) Therefore, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"21\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"52\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId4\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e increases monotonically towards 1 as \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e increases, and can be interpreted as the cumulative probability of completing the game in \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e moves. Or, in other words, the probability of completing the game in \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e moves or fewer.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThis is convenient because there's a result in probability theory that allows you to calculate the expected value of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e (that is, the average number of moves to complete the game) from the cumulative probability. Skipping the math details, the punchline is:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"49\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"136\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId5\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAn infinite sum may not seem helpful, but remember that \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"21\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"52\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId4\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e increases monotonically towards 1, which means the terms in the sum go to zero. Therefore, in practice, you can find the average number of moves to complete the game by summing \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"21\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"76\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId6\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e until the terms are too small to make any practical difference.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven a transition matrix \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eT\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, return the average number of moves \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"19\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"14\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId7\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, accurate to 3 decimal places.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExample\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWith \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e = 8 and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e = 3, and square 5 forbidden, the transition matrix is:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[T = [0      1/3    1/3    1/3    0      0      0      0       \\n    0      1/3    1/3    1/3    0      0      0      0       \\n    0      0      1/3    1/3    0      1/3    0      0       \\n    0      0      0      1/3    0      1/3    1/3    0       \\n    0      0      0      0      0      0      0      0       \\n    0      0      0      0      0      1/3    1/3    1/3     \\n    0      0      0      0      0      0      2/3    1/3     \\n    0      0      0      0      0      0      0      1];\\n\\navgm = avgmoves(T)\\navgm =\\n6.3750]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAn inexact way to verify the result is to directly simulate playing the game many times:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[function avgm = simulategame(n,m,nogo,nexp)\\nmv = zeros(nexp,1);\\nfor g = 1:nexp\\n    % Start game at 1\\n    s = 1;\\n    % Go until we reach the nth square\\n    while (s \u003c n)\\n        % Roll die and see where we would land\\n        snext = s + randi(m);\\n        if (snext \u003c= n) \u0026\u0026 ~any(snext == nogo)\\n            % Move only if valid\\n            s = snext;\\n        end\\n        % Increment number of moves for this game\\n        mv(g) = mv(g) + 1;\\n    end\\n    % Go to the next game\\nend\\navgm = mean(mv);\\nend\\n\\nn = 8;\\nm = 3;\\nnogo = 5;\\navgmsim = simulategame(n,m,nogo,1e4)\\navgmsim =\\n6.3626]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThis approach would require too many simulations to be accurate to the required precision, but gives rough verification that the game takes 6.375 moves on average.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor n = 16, m = 5, and the forbidden squares are 4, 6, 10, 11, and 15, the transition matrix is:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[idx = [1;17;18;33;34;35;65;66;67;69;98;99;101;103;115;117;119;120;133;135;136;137;183;184;185;188;200;201;204;205;217;220;221;222;252;253;254;256];\\nTvals = [0.4;0.2;0.4;0.2;0.2;0.4;0.2;0.2;0.2;0.4;0.2;0.2;0.2;0.4;0.2;0.2;0.2;0.4;0.2;0.2;0.2;0.4;0.2;0.2;0.2;0.4;0.2;0.2;0.2;0.6;0.2;0.2;0.2;0.8;0.2;0.2;0.2;1];\\nT = zeros(16);\\nT(idx) = Tvals\\nT =\\n0.4000    0.2000    0.2000         0    0.2000         0         0         0         0         0         0         0         0         0         0         0\\n0    0.4000    0.2000         0    0.2000         0    0.2000         0         0         0         0         0         0         0         0         0\\n0         0    0.4000         0    0.2000         0    0.2000    0.2000         0         0         0         0         0         0         0         0\\n0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0\\n0         0         0         0    0.4000         0    0.2000    0.2000    0.2000         0         0         0         0         0         0         0\\n0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0\\n0         0         0         0         0         0    0.4000    0.2000    0.2000         0         0    0.2000         0         0         0         0\\n0         0         0         0         0         0         0    0.4000    0.2000         0         0    0.2000    0.2000         0         0         0\\n0         0         0         0         0         0         0         0    0.4000         0         0    0.2000    0.2000    0.2000         0         0\\n0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0\\n0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0\\n0         0         0         0         0         0         0         0         0         0         0    0.4000    0.2000    0.2000         0    0.2000\\n0         0         0         0         0         0         0         0         0         0         0         0    0.6000    0.2000         0    0.2000\\n0         0         0         0         0         0         0         0         0         0         0         0         0    0.8000         0    0.2000\\n0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0\\n0         0         0         0         0         0         0         0         0         0         0         0         0         0         0    1.0000]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eComputing the average number of moves from the transition matrix and from simulation:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[avgm = avgmoves(T)\\navgm =\\n11.4015\\navgmsim = simulategame(16,5,[4 6 10 11 15],1e4)\\navgmsim =\\n11.4266]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eAssumptions\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe input will be a valid transition matrix for a game with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u0026gt; \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e ≥ 2. The output should be a double-precision scalar that has the average number of moves, accurate to at least 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Number of Moves for the Royal Game of Err","description":"Bored of tedious court assemblies, King Neduchadneddar the Procrastinator has found a cunning way to keep his advisors, councilors, viziers, and other courtly lackeys busy: he plays a simple and pointless board game with them.\r\nPlayers move their pawn along a path of n squares, starting at square 1, with the goal of being the first to reach square n. On each turn, they roll an m-sided die to determine how many squares to move. However, if their move would take them beyond the nth square, they stay put. Also, players cannot land on any of the numbers that King Neduchadneddar has decided are the unlucky numbers of the day. If their move would land them on any of these squares, they stay put instead.\r\nTo keep his courtiers on their toes (so they have less time to plot his downfall), the king changes the number of squares (n), the number of sides of the die (m), and the list of unlucky forbidden squares each day.\r\nNeeding to know how much time he will waste with each game, the king in his wisdom (and/or insanity) has appointed you Chief Royal Mage of Matrix Computations and tasked you with determining the average number of moves it will take a player to complete a game according to today’s setup.\r\nYou can use the transition matrix for the game to do this. The transition matrix T is an n-by-n matrix where T(j,k) is the probability of moving from square j to square k in one move. Then  gives the probability of moving from from square j to square k in m moves (where  is a matrix power; that is, , using standard matrix multiplication).\r\n is the probability of moving from square 1 to square n in m moves. That is, it is the probability of completing the game in m moves. However, once you reach square n you stay there forever. (In practice, the game would end at this point.) Therefore,  increases monotonically towards 1 as m increases, and can be interpreted as the cumulative probability of completing the game in m moves. Or, in other words, the probability of completing the game in m moves or fewer.\r\nThis is convenient because there's a result in probability theory that allows you to calculate the expected value of m (that is, the average number of moves to complete the game) from the cumulative probability. Skipping the math details, the punchline is:\r\n\r\nAn infinite sum may not seem helpful, but remember that  increases monotonically towards 1, which means the terms in the sum go to zero. Therefore, in practice, you can find the average number of moves to complete the game by summing  until the terms are too small to make any practical difference.\r\nGiven a transition matrix T, return the average number of moves , accurate to 3 decimal places.\r\nExample\r\nWith n = 8 and m = 3, and square 5 forbidden, the transition matrix is:\r\nT = [0      1/3    1/3    1/3    0      0      0      0       \r\n    0      1/3    1/3    1/3    0      0      0      0       \r\n    0      0      1/3    1/3    0      1/3    0      0       \r\n    0      0      0      1/3    0      1/3    1/3    0       \r\n    0      0      0      0      0      0      0      0       \r\n    0      0      0      0      0      1/3    1/3    1/3     \r\n    0      0      0      0      0      0      2/3    1/3     \r\n    0      0      0      0      0      0      0      1];\r\n\r\navgm = avgmoves(T)\r\navgm =\r\n6.3750\r\nAn inexact way to verify the result is to directly simulate playing the game many times:\r\nfunction avgm = simulategame(n,m,nogo,nexp)\r\nmv = zeros(nexp,1);\r\nfor g = 1:nexp\r\n    % Start game at 1\r\n    s = 1;\r\n    % Go until we reach the nth square\r\n    while (s \u003c n)\r\n        % Roll die and see where we would land\r\n        snext = s + randi(m);\r\n        if (snext \u003c= n) \u0026\u0026 ~any(snext == nogo)\r\n            % Move only if valid\r\n            s = snext;\r\n        end\r\n        % Increment number of moves for this game\r\n        mv(g) = mv(g) + 1;\r\n    end\r\n    % Go to the next game\r\nend\r\navgm = mean(mv);\r\nend\r\n\r\nn = 8;\r\nm = 3;\r\nnogo = 5;\r\navgmsim = simulategame(n,m,nogo,1e4)\r\navgmsim =\r\n6.3626\r\nThis approach would require too many simulations to be accurate to the required precision, but gives rough verification that the game takes 6.375 moves on average.\r\nFor n = 16, m = 5, and the forbidden squares are 4, 6, 10, 11, and 15, the transition matrix is:\r\nidx = [1;17;18;33;34;35;65;66;67;69;98;99;101;103;115;117;119;120;133;135;136;137;183;184;185;188;200;201;204;205;217;220;221;222;252;253;254;256];\r\nTvals = [0.4;0.2;0.4;0.2;0.2;0.4;0.2;0.2;0.2;0.4;0.2;0.2;0.2;0.4;0.2;0.2;0.2;0.4;0.2;0.2;0.2;0.4;0.2;0.2;0.2;0.4;0.2;0.2;0.2;0.6;0.2;0.2;0.2;0.8;0.2;0.2;0.2;1];\r\nT = zeros(16);\r\nT(idx) = Tvals\r\nT =\r\n0.4000    0.2000    0.2000         0    0.2000         0         0         0         0         0         0         0         0         0         0         0\r\n0    0.4000    0.2000         0    0.2000         0    0.2000         0         0         0         0         0         0         0         0         0\r\n0         0    0.4000         0    0.2000         0    0.2000    0.2000         0         0         0         0         0         0         0         0\r\n0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0\r\n0         0         0         0    0.4000         0    0.2000    0.2000    0.2000         0         0         0         0         0         0         0\r\n0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0\r\n0         0         0         0         0         0    0.4000    0.2000    0.2000         0         0    0.2000         0         0         0         0\r\n0         0         0         0         0         0         0    0.4000    0.2000         0         0    0.2000    0.2000         0         0         0\r\n0         0         0         0         0         0         0         0    0.4000         0         0    0.2000    0.2000    0.2000         0         0\r\n0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0\r\n0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0\r\n0         0         0         0         0         0         0         0         0         0         0    0.4000    0.2000    0.2000         0    0.2000\r\n0         0         0         0         0         0         0         0         0         0         0         0    0.6000    0.2000         0    0.2000\r\n0         0         0         0         0         0         0         0         0         0         0         0         0    0.8000         0    0.2000\r\n0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0\r\n0         0         0         0         0         0         0         0         0         0         0         0         0         0         0    1.0000\r\nComputing the average number of moves from the transition matrix and from simulation:\r\navgm = avgmoves(T)\r\navgm =\r\n11.4015\r\navgmsim = simulategame(16,5,[4 6 10 11 15],1e4)\r\navgmsim =\r\n11.4266\r\nAssumptions\r\nThe input will be a valid transition matrix for a game with n \u003e m ≥ 2. The output should be a double-precision scalar that has the average number of moves, accurate to at least 0.001.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 2373.27px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 644.075px 1186.64px; transform-origin: 644.081px 1186.64px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eBored of tedious court assemblies, King \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eNeduchadneddar\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e the Procrastinator has found a cunning way to keep his advisors, councilors, viziers, and other courtly lackeys busy: he plays a simple and pointless board game with them.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 42px; text-align: left; transform-origin: 385px 42px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ePlayers move their pawn along a path of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e squares, starting at square 1, with the goal of being the first to reach square \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. On each turn, they roll an \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e-sided die to determine how many squares to move. However, if their move would take them beyond the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eth\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e square, they stay put. Also, players cannot land on any of the numbers that King \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eNeduchadneddar\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e has decided are the unlucky numbers of the day. If their move would land them on any of these squares, they stay put instead.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eTo keep his courtiers on their toes (so they have less time to plot his downfall), the king changes the number of squares (\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e), the number of sides of the die (\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e), and the list of unlucky forbidden squares each day.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 31.5px; text-align: left; transform-origin: 385px 31.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eNeeding to know how much time he will waste with each game, the king in his wisdom (and/or insanity) has appointed you Chief Royal Mage of Matrix Computations and tasked you with determining the average number of moves it will take a player to complete a game according to today’s setup.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 73.6px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 36.8px; text-align: left; transform-origin: 385px 36.8px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eYou can use the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003etransition matrix\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e for the game to do this. The transition matrix \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eT\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is an \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e-by-\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e matrix where T(\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ej\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e,\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ek\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e) is the probability of moving from square \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ej\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e to square \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ek\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e in one move. Then \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"51\" height=\"21\" style=\"vertical-align: baseline;width: 51px;height: 21px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e gives the probability of moving from from square \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ej\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e to square \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ek\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e in \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e moves (where \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"20\" height=\"20\" style=\"vertical-align: baseline;width: 20px;height: 20px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is a matrix power; that is, \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"84\" height=\"18\" style=\"vertical-align: baseline;width: 84px;height: 18px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, using standard matrix multiplication).\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 95.6px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 47.8px; text-align: left; transform-origin: 385px 47.8px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cimg class=\"imageNode\" width=\"52\" height=\"21\" style=\"vertical-align: baseline;width: 52px;height: 21px\" src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGgAAAAqCAYAAABfjB7GAAAAAXNSR0IArs4c6QAACU9JREFUeF7tWn2MXFUVP+e+menOLpTWaCvrdN57M5NuWJSoMfiBxhqpKW3jAtqaNeAHtTXEQgsSikLVghUhpECojbgBFqi1RRopETEh4CoRqR+RSNymZfbdO+22WhWCFHednTf3OGfznnmdnZl9s195JfP+adPee+6553fPx++cQWh9kbYARlq7lnLQAijij6AFUAugiFsg4uq1PKgFUMQtEHH1Wh7UAijiFoi4ei0PagEUcQtEXL1Z9aDOzs72efPmXQ4AywHgQgC4a2RkZG9HR8c1RHQjEZ2IxWI9ADDsum4vIt5aWXMWEa1USr0YcdvNiXqzChDfgEFKJBJ9iHixEKJXa70SEZ/QWl+OiF8hojWIuEpr/SvDMM4loh8g4jrHcR6cEwtE/JCaAJmmaQshngKA86aof0FrfUmhUDiUSqXeFo/HnyQiRMTfAsDuYrGY90B7DwAcJKLHLct6tlAo7ACAzyHiasdx/lDv7FQqlUwkEpuIaEhK+dN66yzLsoQQWyq6HFRK9U/xLjO5zbBt+2oAeMM0zT0DAwPuZMJrApTJZFYR0c8B4PmKVbfF4/HfHz58+NSyZctinhGvYcFEdLtS6mb+ayaTOYeIrgWArQAwQERrlVKvm6b5PiHELwFgIXuLUurJVCrVGY/HWf75AHC1lPKhVCq1wAOy6LrumuHh4ddqKb9kyZJOwzDuQ8QDUspH+ezqdQwMIt4AAOsAoC1KHsk2PHr06AYi6jIMY2s+n3+jEUi1AELLsrYj4ke01r2FQuFvvgDLshYg4mNeTgHvpbOnjX/e/z9KRC8HgLuKiB4AgL5isbj5xIkTI5ZlfRwRn0HE/W1tbesHBwff9IFExH7Hcb4BALpacdu2uwDgIUTscxyHPeI0cNLp9ELDMDYioqG1TiPil1lGlABifTyQthKRlUwmv8b3rwfSBIACRt5enagD3rCoEooOaa1XFQoF6Qv3840QYo/jOE/lcrl5ruvuQsQ1ALBaSvkbXmvb9jcrf34HES/jdfxvmUxmHEgiulQpdaBaYdM0zxVC/ISIXrAs61u1wgNffGBggIHVuVzu/HK5/DQALIkaQHy37u7us0ZHRznfqnQ6fVu9cDcBoGw2u4SIrhBC7Mjn88WgoXwjeuFtz9jY2Hr2iIAHtQkhtpXL5R8ycIFcNlTJF1dyyOvq6jp7bGyMQ9M7PTD+3t3dnRgdHd0JAOxZq7XWna7rvuyHOQZaa30vEV0ohOgZGho6NlnsNk3zPCEEA2RGESDW3zTN9wshngCAW6SUj9S6U+gqLmDE9R5Am5VS9zYylJ/LEPF6x3Hu4ZBk2/YFAPA0Iu72Q1k2m12ktWZPKhARh75RKeVuP8x5cn6GiJsdx9k1GTje5SMPkJfT7wCAT9R7eKEByuVy7yiXyxx6PgwAp4houVLqYANj+blsXbAqqxXKbNs2iWg/Ii5CxJvS6fRjvstzKBgZGXkYET+qtV5RKBT+/FYByAvtfkF2rZSSo8hpeTU0QJZlfZATOwCcDQAvlUql1cPDw8fDGGs6a7LZLBcrXAW+6FeGYeTNcojDVCq1MJFIdDMB11pfxIVJqVS6isNyOp3OGIZxEwB8FgDeRMTrHMfZX0vvQBp43TCMnnw+/8/gumYA2oSIHKb460smkxsHBwfHwhhrOmu8gmJ7s2fOJkC5XG6+1no5EfUCwGe8+3Eeud227VUAwGHrTwDAf18IAM/Ue1yByniZF5V+3TRAlmUxl7gfAL7Am+cq6S5evLijvb2dS3Qmr3c4jsOvMtQ3mwAFiiIuagYAoMTGFULEiOgGIcSG0dHRVz0y/nkAOJBIJK5kLlmtfKDTwutullJ+r2mAUqnUuzxi+V4A+EczuSCUNessWrp06dtLpRITWg6vTbV/5gggP6r8DhGZ12zQWm9m7uh3UCrec5FXpbHhJ5Bqfu+2bd9ZCYVMrB8hoq8qpf7rmyRUiPOJJQDEG7nrdMCotTeTySzVWv8CEbNRAyj48hFxLxEtqLSztvncMcDDmE5wQXVa6AreN5PJfJ+IttSybSiAAnngtPbOTANSLW86XjCdvWHuVdWv5NB1nZSSG7zjXmLbNhcI3Cf8o+u6PceOHTtRT24AoIPxeHz1kSNH/hXag4J5wMs/3Mj8f3snzGWmumY6Rp7O3jD6BvqVnJP3+i0r3lvVs7zPNM3rGzVGpwVQ1UuZ0N4Jc5mprgkQ2A9ELMT5HI97hqeEECuGhoZe8O9ZxRnXSCkfb2SDAEATiolJQ1zwpVTGD/tGRkbWnTx58j9TNXoz+wJtIR7qfVdKyZ3yUN9selCwaUxEtVpePmd8zR+71FO6ytsm0JfJAAq+FD5jQhkYylpTXxQ8/y4p5Y11KqEJJzQJEJ9zDgtRSv17sjOCTWPmQkqpvUEFLMvyq7txj3BdN1kh2x+SUnJFelqXPlhsENGE9llDgGqMFy52HOfZqdu7+Z0BD67LJWpJbQIgzGQyXyIi5nlcpT6QTCY3NxoBBJrGRwzDWJnP54d8Hap5jWEYO8vl8m2GYfwon8//tVrXQDl+QXWoHM/5DUzGr+rTiMiVCCvOyXDOAeIBXSwW4x5gR/V4oxHclmX18GjdW8MsvyYPqTIoLx8nnfXK4mDTuFZ4q+Ju/Vrr+UKIXfUetl+OE9ErtQaVEwBihdva2r5IRNym+JQPjnfRPA/ZAOCIEOLH1eOI5v0j1A5+4Vt4egsADROuN39aAQBrEZF/rNLmncDEj0ngc7FY7LmqfhcTxSsqLRnuzHNbpmGnJEjaaxUuXk5hTvNtIvoLd/KllM/XC5u+N9YrgibLQaEsONuLeEaltT5ARIeqZ1AzeTZTimQy2QcA9zciljN1Jv9MoDKI7iei9lKp1FtrzH9GAMQGsW2bp7L9iLh2tnhYNpt9NxF9HQB47sTFwqx+3p12ElFPvZ+ZnTEABeb4n3Rdd20jZj4Vq3oj9XsMw7i1VjKfisxGezj3uK67DxHvrPfjl8mKhJnWadryOMeUy+VbKtPYXLFY3Hj8+PFXpy0UAHh+I4TgLvTdjuO8MhMyG8nwHsODRPSwUmpfo7L+jPEg/8LsSUopntJ+bGxsbNNMgTTboPjyuTNjGMYOIrq7UfHgrz/jAPIV9179fKXUS3Nl3Bk4B70JcV5KeTKMvDMWoDCXeyusaQEUcRRbAEUcoP8Bgqk+dnshbtUAAAAASUVORK5CYII=\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the probability of moving from square 1 to square \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e in \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e moves. That is, it is the probability of completing the game in \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e moves. However, once you reach square \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e you stay there forever. (In practice, the game would end at this point.) Therefore, \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"52\" height=\"21\" style=\"vertical-align: baseline;width: 52px;height: 21px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e increases monotonically towards 1 as \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e increases, and can be interpreted as the cumulative probability of completing the game in \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e moves. Or, in other words, the probability of completing the game in \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e moves or fewer.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 31.5px; text-align: left; transform-origin: 385px 31.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThis is convenient because there's a result in probability theory that allows you to calculate the expected value of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e (that is, the average number of moves to complete the game) from the cumulative probability. Skipping the math details, the punchline is:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 54.8px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 27.4px; text-align: left; transform-origin: 385px 27.4px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cimg class=\"imageNode\" width=\"136\" height=\"49\" style=\"vertical-align: baseline;width: 136px;height: 49px\" src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAREAAABiCAYAAAB3VedVAAAAAXNSR0IArs4c6QAAF05JREFUeF7tXXuYHFWVP+dWzySTUR6u8ggz3VU9YyIRdXdFdBGXiKBAYHkGiMiCYGAFsgkhGqK85BF55NtPN8LuGiGBjYCIQHjqsrBRVoQPdgE/CV/yzVRVJ0N47AosmJnMdPe92yfewpua6u6q7ulJ98yp/2b63Fu3fufWr+49r4vAFyPACDACdSCAdbTlpowAI8AIAJMITwJGgBGoCwEmkbrg48aMACPAJMJzgBFgBOpCgEmkLvi4MSPACDCJ8BxgBBiBuhBgEqkLPm7MCDACTCI8BxgBRqAuBJhE6oKPGzMCjACTCM8BRoARqAsBJpEE8DmOkwGAywDgdN3sYaXUNb7vv0B/z549O+X7/skAsAQRPwkAmwDg2kwmc8f69esLCW7FooxAyyDAJBJTVb29vR8tFotrAeAHQohfSikXAcC5JcL4AyLOdV33CcdxlhBpAMAA/R8APlL6u00pdYVt28uZSGKCzWIthQCTSAx10Qojl8tdDwBvep63HAAUAAjHcb4KAP8EAB4ifk8ptQARl7iu+3MAkNls9sNKqdUA8CHLso7u6+vrj3E7FmEEWgoBJpEY6urp6dlLSklblyW+7/8yaELksnnz5suUUpcDQB4RT3Bd92Gzy2w2O0cpdV/UbzFuzSKMQNMjwCQSQ0WZTGZ/IcSjiHhBmCS6u7unp1KpuwHgILKVeJ73U7PLrq6u/dra2h5CxJWu694a43Yswgi0FAJMIjHUlc1mZ0gpHxFCPJROp5eYto1Zs2a9b/v27auUUqeRIdWyrBP7+vpeCrqdMWPGB/P5/ANKqWXmKibGbVmEEWgJBJhEYqiJiGJwcPA2RPw8Ii5wXfdOsnk4jrM3ANwAAPsCwDsAcBIArBNCnNvf3/8Gdd3T03OwlPKyfD5/+sDAwJsxbscijEBLIcAkElNdjuPMBYAfk7eFDKmB90UptUlKeQoiviWEWAMAXwSAJxHxO4g4JKVcjohXu677eMxbsRgj0FIIMInEVxfatk0Eca2OAXkNAFaVPC83eZ73OnWz9957d3Z2di5QSi0EgD0A4HEiE9d1n9Menfh3Y0lGoEUQYBJpEUXxMBmBZkWASaRZNcPjYgRaBAEmkRZRFA+TEWhWBJhEmlUzPC5GoEUQYBJpEUXxMBmBZkWASaRZNcPjYgRaBAEmkRZRFA+TEWhWBJhEmlUzPC5GoEUQYBLRispms19QSlHy3J7jpTtEPIeT8sYLbb5PoxBgEtHIhtL6G4X3Tv0yiYwLzHyTBiPAJGIAnM1mdweANUqp4yNw36SUulcIMZJQJ7sppf4KAD4BAFPNtkwiCZFk8aZEgEkkpBZdBvFeAJgR+omKDp3nui4l2VFls0SXJqhlSqnFOokPmEQSQcjCTYoAk0iEYhzH+VsA+FHwshsio+qFJNFreMvEJJIEPZZtVgSYRCI0o6u2X4WIy8I/I+L9AHCW67r/V4tSu7q6PtDW1kb1SL7IJFILgtym2RBgEimjkUwms68Qgl72QyOI5Kp0On11rdXbbds+DRHvZBJptteBx1MLAkwiFVDLZDJ/KYR4EACmh8Te0sdE1FRoKJPJOEKIhxFxBbt4a5m23KaZEGASqawNzGazZyml/iVsH1FKvZRKpY6r5RgIKl40bdq0WxDx35hEmul14LHUggCTSBXUpk+fPm3q1Kk3KaXOihC9vaOj44INGzbQQVWJLsdxrkZEj0kkEWws3IQIMInEUIpxLMRnI8S/5nkeHQWRyO1LWxrqK5fLUb1WvhiBlkWASSSm6iqExb+llDqBj4OICSSLTTgEmETiq5TsI0uVUt+NaEJn887L5XKvxu+OJRmBiYEAk0gCPdL5M0NDQzcBAAWj7XQRudi2fXmtbt8Ew2BRRqCpEGASSaiOSmHxAED2kdsTdsnijEBLI1CWRHp6eg5QSu1Ty9MVi8X+iWwwDA7pHuuw+Fqw5jaMwK5GoCyJZLPZ65RSS2sZ4ESPxKxUNqDesPha8OY2kQiInp6ez0gp6WhT8qr1tbe3f314ePhgRLwRAD6iD2C/J5vNHiilvA4RD6NUB9d1r0/qbZvMOuDtTI3a7+np2atYLFLo+mHhLhCxrrD4GofEzSIQyGazZyulKLDvYinlm0KIDgDYopSiSOTLycsOADMR8VGl1B1KqSdHRkbmb926dZABjYcAk0g8nCKlaMsnpXwIADIhgbrC4uMMSScJfgUAdvd9/wcAUIxqN2PGjA+OjIycj4hTPM/7dpy+d7WM4zjfouNKax0HGbl936dnVRTUBwALlVI3IeLLnuet1VHI/6yPQX3VsqwbpZSHK6UeQsTFrut+r9xKpKurq6O9vZ366/c8jyrhRV62bdtCiKVSymd836fyEbv06u3t7SkWi1SG4gbP84g4I6+4cmZjJpH6VFspLP4pADjJ9306s3dMr97e3t2KxeLVSqnf27a9PMojROSRz+f/DgBo4uyJiNe7rnvJmA6kAZ3NnDnz/SMjI/8KAF8iMrAsa3V3d/er9IyZTOYvhBA/B4C9aAVRekGPyuVyLwOAsG37IET8PgAcBABzPc+7x+jrOABYaVnWN7q6uoq5XO4fAGABAKy1LOuCvr6+dwKyEUIc2d/fT7obdVHQoWVZKxFxned5NMZRAYZEHoi4pLTaOYeKUDXT1l7nbP2jUmqF7/u/KkeUceUCgJhE6nwRtH1khT7EO+iNChid4Lruw3V2P6o5baOklD8s9f9iVCZxb2/vlEKhcB4A7IuIbQDw92QAbhUSyWQy+5cOTH9QCHFtuACU4zgnA0Dw9V/X3t5+xsaNG98NQLJt+zgikoBcqC8hBG1T/iCEONF13U29vb0fKhaL6yipUghxTH9//+8MstmHqtpFEb/jODNLK87ViLgqqjBVOp3e07KsCxHRklKmEfGrNK5mIhEaj/YuriWic123bAJpXLkdzzjWk3wy9uc4Dn39Hil97f5MT5yG2ESCOBVEbJ86der8Mjk7ltZBURMOEdmBrUIiRBSIeLgQYmFfX99wMJ+IrI0VBP37257nLTfnm23bhyLiRQG5BCUXlFJX+r5P2xpp2/anEfExRLw1nU4v0SucHWQDAA9nMpmF4ZVdUBZCKfVUuVggGt/69esl3UO/gNRfd7ORCOGlo69XWJb1lb6+vpfKvbNx5ZhE6mQ9s8iQ7qrmpLwqQwkiZhdIKY/N5XL/XW3oektDNht6cVpiO+M4znxEfMF13WfN5zNWEFSvNq+UOiKcatDT03NwsVg8xPf9G2bNmtU+NDREtiJanRwT9Gfb9kLyziDiUcGXOFjhKKXmCSH+i+xHtELRX+4pUsrvK6UOEkIc19/fv6Ua7sEKiGxlzUgiwepZSvkxy7Lm9ff3vxH1THHlmESqzYgKv0e4eusqn1hpKEZtk7symczSOJGxrUgi5TAIVhAA8H4AeCGfzx8zMDDwSjn5rq6u/dra2ohAB0pG0DN833+bVnKDg4O3lbYk+xnbFnQc5wYAOBURL6EXK5VKfZfsJPqrPUcpdR8iLnJd9+Y406XZSYSewXGcj5c+LrRaus7zPCLbyATSOHJMInFmRbRM2KjaMI+MXspT7MJi/VWNZWuZSCQSuGq1KlZ1dHRcuGHDhrKV9/XWhrYtSwNvi/Fyv7dt0SsWihs5t2SwvTmfz187MDDwJt3HIJ1DpJRH5nK55+NMl1YgEcMOlK60woojN4pECNSRkZHuQqFwYMkSTsE6n0PE2zzPW0lW8Ewm86WS8YhqYRCTre7o6LhY783ppTpQKXUNAHweAO4XQlxYbqkURxnNLBMOf29kkJJ2u5HNpSilnBM3GniikIixNZlPc0Iptcj3ffLENPSi7ZGUkrxBTyulTqHVTJwbNpJEot5PALjR9/2fkFFdSnmilPJiej/pMDZyR2/atOl/o8YduNKrbbmqyY0iER3uPlMp9U3tLnuX9p/Tpk17aWhoiHz3nwaAgo4CJC/EYel0+mnf98ml9Q0AeBERj9Uh4TtcbXGAD31p4jSJlFFKzW50Wn5E/dVG2UF2PKPhlXgsyWSeKCRibE3+HAB2zEff95+peZLEbGjEq1Rd+ZhdNpJEHMeh84vm6JUTxSfRFvrooaGht9vb28l1vQ955ZRS9CGnq+w7aNiC7qgUYFdNrtx2Jtgnkr/7McuyTi8UCucDwFAqlSIX2iFKqX8HgDfInWZZ1ieklEfk8/kFqVRqL23ppgeccCQSUQn+14VC4ZQtW7ZsjTk3k4q9pwuKqEwSTTlRSCSbzdJ8e0J/mH5jWRaVpfyfpEAmkQ9KWGpbSSKjdCNJhJ6Bqu21t7evQsQv05ywLOsiKSXZdZ70PG+1bdu0e1hPspW2v3rLR3IvV1rhVpOLJBEzSIes+sR2Uspu27avJYOetnBTVN9vaB8JALMty1pMxiidnEYGLTMYKIn+mll23OwgAQjmhEnqYZkoJGLMN4JlZSaTWRzHsFzPRAphl+jM5EaTiLkyU0pRQOE+QojNQfyK4zhnlDy5lE2+Y5VSrg6w4Yqm1cthruv+ZxRm1eQiSSSbzc6QUj6CiD1KqVsRsdjR0bGYbB+h/SkF7bwrpfymLsiDRDT6vJZRwUD1KLUZ2oaqv9d1Il7c59Eu5Ado+7irSKSeZEz9nDV/UEwSrbY8j4tpHDnzHahmMwj312gSCYzGpS0NrX4fRMSUEGKRjquJvXI1x1lp11BNrhyJkFuLVhP5kp/9eSHEyYF/PLQ/3ckjEQozHhUMFEd5zSoTtoMg4prt27df0OhErXpWE/W0NfWwK0kkOF4DAPYfz9VtPURQT9s489+w1dD7+VShUPhysJ22bXsPRLwbAI5AxAsquaXjjrOaXBSJmKsJGuTpZqKR6a8PV/MybjY9KhgoDkDNKBNR8X3MyiEakZjP6nyMnSCohwjqadssetBRk2R/o2vcVrfVXpxK+NTTthruoQ81GZnn+r7/i6CdkV8E1dzSccdZTW4UidDyOZVK/VTXVhh1ZKSxP6Wl1FGe5/02eADDi1A1GCgMVhN7Z8J2kK1xI0arTQj6XUdiUjDTJVF70noMfBOBREIZveO2ug2lDDSNTcR8ocnUkEqlzjfTA4z3qKonL0QOFNUbGX9UTW4UiZgrjfBeMLQ/3cntFcptSOQSo5epWUkkVOV9zO0gtm0fCQCUwEeh7KOOjwjhertS6jzf97fHIahWJ5HQV5c8DYdXShqLg0lcmdC9r/E877K4bat9ueP2EyVnfKhHhf6b9kqzHEK5+xmrlo5KbvNqclEkQrkF5HkZZdk1gp5mhF1Hob0YMfdqx3GOoH2s53kb6wFuV7WNsIOMaWKdzri9GRG7K8V/GAT7k8HBwXNef/31bXEwaXUSCRn0Kroh4+CRUMbc1q/wPI/ipmKdLZSQRERXV9cenZ2deTMjOWqs5geFXN6FQmFuEF1L8qa9Ur+fj+h3cGNUDRHDdVtx51BNbicSCfmfRy2VDBYcpdAwWwkh3kcvxvDw8EWNNj4mnByxxCMqu4+ZHSQYgG3bFP1Lqe13VQrjNrDdWi1nxHy4VieRUOp/IgKNpeQqQka4QiJbTFwSCeVe0crimnL1YYytL3lEKQnxUp3F/B6xGV6bPsuyji0UCr2lsgqfzGQyN0S5xANXcNS2yISmmtxOJBLab83zff+uoLMQC44KejIUThmB1K5zZGRk6SuvvPL7sVDoOPdBbrKzS+T+I33fsU6so6/cX1M6AWV6VgvjNnI4jkgSrZnNZj+lvWx7VZso44xv1dvp+hyrKQtXC487ieiTD+ml7UySbqDrmpA9ka5RL3vw8CbJ6/9VdIUbpoapUY4LY8WaQ8S7lFIfCEIzIgA3XcE7vesh2apyO5GIQQSjHsYAlPJjRuUu6AfcARwVhtm2bdvKuMvuqjNqnAXCdpCwh6rW4dCeddu2bQcIISj693SqfKXxKmvUCu7lOM5cAPhx6e8rw3U0zPHoiNpDEJEKFNM99tS/k6ftXkT8RbFYXB83/6bWZ621HcXiWJZ1hlKKxt9t9LNj/KV4hpeEEPcFqfq13idmO/PAsorR13prSvatUxDxxEC3AED2K7JlPZFKpZ4wI231h5mKoV8ayFdK2zCDPKOidnVsC+l4egm7WyzLujrIRg4/r1FagVZAp5arwBdHjrN4Q+iS3adQKKwrbTM+GnOi1StGqQNVM0Sz2ezuALCGvi6VlF7vYLj9zgj09PR0SynXlTB/OUnKQVIctTfodinlRbrkY9IuEsnrrRrFk5xVqVZsHDkmEQP6SifcJdJQMuHnhBBz4mQ727b9GUSkbN4rKtWASHZ7lq6GgF4FrkHEUxpR8pLur+1jh2cymWWNDukPCmkhoheuIGdiEVeOSUSjprcB30LE71SbVGP8e5K9/o6YFSnlty3LOn6clvRj/Lit151hAP1CI5ItdW7K5VLKRY0+z9l4FirzcVa5+8WVI20yieg5HbKDjNtMT5oPQzVdHMehIsBnSinPbFbbxrgBOE43IptHsVi8FBF7h4eHLxwrh4E2ftPRH8s9z3u9wY+z4yMEAH+DiOdVWP3GldsxXCaRP1XAJqPdjAYrcVT3SZO7dAdkMT+mVIrhfCnl+Uwk46M1vVo9BxE/NzIysnCsiGQ8Rq9XFvOllB9OpVJXljO4xpUzx8wk8scv+wGISGeZjPslhHix1toYjuPsLYTo1eekxAqEGvcHnIA3TKfTWSHEbr7vv9Aqj0eGW/pe9vf3P00V6cuNO64ck0iraJ7HyQi0AAK8EmkBJfEQGYFmRoBJpJm1w2NjBFoAASaRFlASD5ERaGYEmESaWTs8NkagBRBgEtk1SgrO6Lmi5OLdIqXc7Y8pR7gsKmV71wyR78oIxEOASSQeTmMqpQPb7tDVzNbMnj3b2rx582VKqdMsyzqx0iHLYzoQ7owRGAMEmETGAMQkXQTZ0Eqpd8yiMkHBJ6XUc41M9EoyVpZlBOIgwCQSB6UxlLFt+zREvJNKIprVsoxaqkcnqRkyhkPjrhiBmhBgEqkJttoamYWdosLdg6MZqhUpqu3u3IoRaAwCk5ZEqBTklClTqHgM1YE9iFYGg4ODd3V2di6gc4iVUltTqRRV1RooFArzEPGqkgyVfDza930KHU58hU4WHFWIyCg6My6nvCV+AG7ACEQgMGlJhLAwasoeLoSYJ6U8GhHvp5PVEfFrdKYHIs6RUv6HZVn7KqVuohVEKpV6IJ/P0+FedLh5rIuydVOp1IqgXVQFq6C8XdIzd2MNgIUYgQYhMKlJJDiiUilF7tVflxKU1g4PD/fpw5I/VjpX5xml1D2lA5Ifz+VydOL6qVRFWwjxWynlp4wSeFXVg4ivWZb1GpNIVahYoMUQmNQkYlRR31OfJPZAV1fX9La2NlplUHnEr9Mp61TSv62t7QGl1HC4TH8SfVfbzjiOM7+0QPrheB1anWTsLMsIlENgUpOIUR171fDw8CI62iIou4+IP5s6dep8OsQ8IBs6f9d13WWVUqkrTTU2rPKLOBERmLQkYhwcRVXUj/E871ekYH1s45WIeEJQT9OwVRzv+/66iFL/VedGUMEsqKgfrmhmuniFEEfqGiFV+2UBRmBXIzBpScQ4bb5fKXWG7/tvG9uNfZRSRBivGUcTHkr2ECkllePfSMWAktpEqCZqjGCz302bNu1MWgHt6snB92cE4iAwaUkkON0MERe7rkvHhirHcT5e8rg8WqqjuTbYthgHO+eUUo8h4pDneWtr3dKQUrLZ7ElKqdWIuNB1XTPsneqmnpDL5Z6PozyWYQSaAYHJSiLBOatUL5PiNZ7VL/fZSqlb9CqETj6j7Q2dUPczKp9IuS7pdPruMSjpTwl4n1VKXYaIbpCAJ6W8dPPmzW4zTAweAyMQF4HJSiJx8WE5RoARqIIAkwhPEUaAEagLASaRuuDjxowAI8AkwnOAEWAE6kKASaQu+LgxI8AIMInwHGAEGIG6EGASqQs+bswIMAL/D8gJFzWb99mgAAAAAElFTkSuQmCC\" data-image-state=\"image-loaded\"\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 74.6px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 37.3px; text-align: left; transform-origin: 385px 37.3px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eAn infinite sum may not seem helpful, but remember that \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"52\" height=\"21\" style=\"vertical-align: baseline;width: 52px;height: 21px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e increases monotonically towards 1, which means the terms in the sum go to zero. Therefore, in practice, you can find the average number of moves to complete the game by summing \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"76\" height=\"21\" style=\"vertical-align: baseline;width: 76px;height: 21px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e until the terms are too small to make any practical difference.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 24.8px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 12.4px; text-align: left; transform-origin: 385px 12.4px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eGiven a transition matrix \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eT\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, return the average number of moves \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"14\" height=\"19\" style=\"vertical-align: baseline;width: 14px;height: 19px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, accurate to 3 decimal places.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExample\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWith \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e = 8 and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e = 3, and square 5 forbidden, the transition matrix is:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 245.25px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 641.075px 122.625px; transform-origin: 641.081px 122.625px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eT = [0      1/3    1/3    1/3    0      0      0      0       \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    0      1/3    1/3    1/3    0      0      0      0       \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    0      0      1/3    1/3    0      1/3    0      0       \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    0      0      0      1/3    0      1/3    1/3    0       \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    0      0      0      0      0      0      0      0       \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    0      0      0      0      0      1/3    1/3    1/3     \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    0      0      0      0      0      0      2/3    1/3     \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    0      0      0      0      0      0      0      1];\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eavgm = avgmoves(T)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eavgm =\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e6.3750\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eAn inexact way to verify the result is to directly simulate playing the game many times:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 551.812px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 641.075px 275.9px; transform-origin: 641.081px 275.906px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"border-block-end-color: rgb(14, 0, 255); border-block-start-color: rgb(14, 0, 255); border-bottom-color: rgb(14, 0, 255); border-inline-end-color: rgb(14, 0, 255); border-inline-start-color: rgb(14, 0, 255); border-left-color: rgb(14, 0, 255); border-right-color: rgb(14, 0, 255); border-top-color: rgb(14, 0, 255); caret-color: rgb(14, 0, 255); color: rgb(14, 0, 255); column-rule-color: rgb(14, 0, 255); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(14, 0, 255); text-decoration-color: rgb(14, 0, 255); text-emphasis-color: rgb(14, 0, 255); \"\u003efunction \u003c/span\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eavgm = simulategame(n,m,nogo,nexp)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003emv = zeros(nexp,1);\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"border-block-end-color: rgb(14, 0, 255); border-block-start-color: rgb(14, 0, 255); border-bottom-color: rgb(14, 0, 255); border-inline-end-color: rgb(14, 0, 255); border-inline-start-color: rgb(14, 0, 255); border-left-color: rgb(14, 0, 255); border-right-color: rgb(14, 0, 255); border-top-color: rgb(14, 0, 255); caret-color: rgb(14, 0, 255); color: rgb(14, 0, 255); column-rule-color: rgb(14, 0, 255); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(14, 0, 255); text-decoration-color: rgb(14, 0, 255); text-emphasis-color: rgb(14, 0, 255); \"\u003efor \u003c/span\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eg = 1:nexp\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    \u003c/span\u003e\u003cspan style=\"border-block-end-color: rgb(2, 128, 9); border-block-start-color: rgb(2, 128, 9); border-bottom-color: rgb(2, 128, 9); border-inline-end-color: rgb(2, 128, 9); border-inline-start-color: rgb(2, 128, 9); border-left-color: rgb(2, 128, 9); border-right-color: rgb(2, 128, 9); border-top-color: rgb(2, 128, 9); caret-color: rgb(2, 128, 9); color: rgb(2, 128, 9); column-rule-color: rgb(2, 128, 9); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(2, 128, 9); text-decoration-color: rgb(2, 128, 9); text-emphasis-color: rgb(2, 128, 9); \"\u003e% Start game at 1\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    s = 1;\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    \u003c/span\u003e\u003cspan style=\"border-block-end-color: rgb(2, 128, 9); border-block-start-color: rgb(2, 128, 9); border-bottom-color: rgb(2, 128, 9); border-inline-end-color: rgb(2, 128, 9); border-inline-start-color: rgb(2, 128, 9); border-left-color: rgb(2, 128, 9); border-right-color: rgb(2, 128, 9); border-top-color: rgb(2, 128, 9); caret-color: rgb(2, 128, 9); color: rgb(2, 128, 9); column-rule-color: rgb(2, 128, 9); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(2, 128, 9); text-decoration-color: rgb(2, 128, 9); text-emphasis-color: rgb(2, 128, 9); \"\u003e% Go until we reach the nth square\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    \u003c/span\u003e\u003cspan style=\"border-block-end-color: rgb(14, 0, 255); border-block-start-color: rgb(14, 0, 255); border-bottom-color: rgb(14, 0, 255); border-inline-end-color: rgb(14, 0, 255); border-inline-start-color: rgb(14, 0, 255); border-left-color: rgb(14, 0, 255); border-right-color: rgb(14, 0, 255); border-top-color: rgb(14, 0, 255); caret-color: rgb(14, 0, 255); color: rgb(14, 0, 255); column-rule-color: rgb(14, 0, 255); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(14, 0, 255); text-decoration-color: rgb(14, 0, 255); text-emphasis-color: rgb(14, 0, 255); \"\u003ewhile \u003c/span\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e(s \u0026lt; n)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e        \u003c/span\u003e\u003cspan style=\"border-block-end-color: rgb(2, 128, 9); border-block-start-color: rgb(2, 128, 9); border-bottom-color: rgb(2, 128, 9); border-inline-end-color: rgb(2, 128, 9); border-inline-start-color: rgb(2, 128, 9); border-left-color: rgb(2, 128, 9); border-right-color: rgb(2, 128, 9); border-top-color: rgb(2, 128, 9); caret-color: rgb(2, 128, 9); color: rgb(2, 128, 9); column-rule-color: rgb(2, 128, 9); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(2, 128, 9); text-decoration-color: rgb(2, 128, 9); text-emphasis-color: rgb(2, 128, 9); \"\u003e% Roll die and see where we would land\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e        snext = s + randi(m);\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e        \u003c/span\u003e\u003cspan style=\"border-block-end-color: rgb(14, 0, 255); border-block-start-color: rgb(14, 0, 255); border-bottom-color: rgb(14, 0, 255); border-inline-end-color: rgb(14, 0, 255); border-inline-start-color: rgb(14, 0, 255); border-left-color: rgb(14, 0, 255); border-right-color: rgb(14, 0, 255); border-top-color: rgb(14, 0, 255); caret-color: rgb(14, 0, 255); color: rgb(14, 0, 255); column-rule-color: rgb(14, 0, 255); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(14, 0, 255); text-decoration-color: rgb(14, 0, 255); text-emphasis-color: rgb(14, 0, 255); \"\u003eif \u003c/span\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e(snext \u0026lt;= n) \u0026amp;\u0026amp; ~any(snext == nogo)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e            \u003c/span\u003e\u003cspan style=\"border-block-end-color: rgb(2, 128, 9); border-block-start-color: rgb(2, 128, 9); border-bottom-color: rgb(2, 128, 9); border-inline-end-color: rgb(2, 128, 9); border-inline-start-color: rgb(2, 128, 9); border-left-color: rgb(2, 128, 9); border-right-color: rgb(2, 128, 9); border-top-color: rgb(2, 128, 9); caret-color: rgb(2, 128, 9); color: rgb(2, 128, 9); column-rule-color: rgb(2, 128, 9); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(2, 128, 9); text-decoration-color: rgb(2, 128, 9); text-emphasis-color: rgb(2, 128, 9); \"\u003e% Move only if valid\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e            s = snext;\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e        \u003c/span\u003e\u003cspan style=\"border-block-end-color: rgb(14, 0, 255); border-block-start-color: rgb(14, 0, 255); border-bottom-color: rgb(14, 0, 255); border-inline-end-color: rgb(14, 0, 255); border-inline-start-color: rgb(14, 0, 255); border-left-color: rgb(14, 0, 255); border-right-color: rgb(14, 0, 255); border-top-color: rgb(14, 0, 255); caret-color: rgb(14, 0, 255); color: rgb(14, 0, 255); column-rule-color: rgb(14, 0, 255); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(14, 0, 255); text-decoration-color: rgb(14, 0, 255); text-emphasis-color: rgb(14, 0, 255); \"\u003eend\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e        \u003c/span\u003e\u003cspan style=\"border-block-end-color: rgb(2, 128, 9); border-block-start-color: rgb(2, 128, 9); border-bottom-color: rgb(2, 128, 9); border-inline-end-color: rgb(2, 128, 9); border-inline-start-color: rgb(2, 128, 9); border-left-color: rgb(2, 128, 9); border-right-color: rgb(2, 128, 9); border-top-color: rgb(2, 128, 9); caret-color: rgb(2, 128, 9); color: rgb(2, 128, 9); column-rule-color: rgb(2, 128, 9); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(2, 128, 9); text-decoration-color: rgb(2, 128, 9); text-emphasis-color: rgb(2, 128, 9); \"\u003e% Increment number of moves for this game\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e        mv(g) = mv(g) + 1;\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    \u003c/span\u003e\u003cspan style=\"border-block-end-color: rgb(14, 0, 255); border-block-start-color: rgb(14, 0, 255); border-bottom-color: rgb(14, 0, 255); border-inline-end-color: rgb(14, 0, 255); border-inline-start-color: rgb(14, 0, 255); border-left-color: rgb(14, 0, 255); border-right-color: rgb(14, 0, 255); border-top-color: rgb(14, 0, 255); caret-color: rgb(14, 0, 255); color: rgb(14, 0, 255); column-rule-color: rgb(14, 0, 255); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(14, 0, 255); text-decoration-color: rgb(14, 0, 255); text-emphasis-color: rgb(14, 0, 255); \"\u003eend\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    \u003c/span\u003e\u003cspan style=\"border-block-end-color: rgb(2, 128, 9); border-block-start-color: rgb(2, 128, 9); border-bottom-color: rgb(2, 128, 9); border-inline-end-color: rgb(2, 128, 9); border-inline-start-color: rgb(2, 128, 9); border-left-color: rgb(2, 128, 9); border-right-color: rgb(2, 128, 9); border-top-color: rgb(2, 128, 9); caret-color: rgb(2, 128, 9); color: rgb(2, 128, 9); column-rule-color: rgb(2, 128, 9); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(2, 128, 9); text-decoration-color: rgb(2, 128, 9); text-emphasis-color: rgb(2, 128, 9); \"\u003e% Go to the next game\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"border-block-end-color: rgb(14, 0, 255); border-block-start-color: rgb(14, 0, 255); border-bottom-color: rgb(14, 0, 255); border-inline-end-color: rgb(14, 0, 255); border-inline-start-color: rgb(14, 0, 255); border-left-color: rgb(14, 0, 255); border-right-color: rgb(14, 0, 255); border-top-color: rgb(14, 0, 255); caret-color: rgb(14, 0, 255); color: rgb(14, 0, 255); column-rule-color: rgb(14, 0, 255); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(14, 0, 255); text-decoration-color: rgb(14, 0, 255); text-emphasis-color: rgb(14, 0, 255); \"\u003eend\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eavgm = mean(mv);\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"border-block-end-color: rgb(14, 0, 255); border-block-start-color: rgb(14, 0, 255); border-bottom-color: rgb(14, 0, 255); border-inline-end-color: rgb(14, 0, 255); border-inline-start-color: rgb(14, 0, 255); border-left-color: rgb(14, 0, 255); border-right-color: rgb(14, 0, 255); border-top-color: rgb(14, 0, 255); caret-color: rgb(14, 0, 255); color: rgb(14, 0, 255); column-rule-color: rgb(14, 0, 255); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(14, 0, 255); text-decoration-color: rgb(14, 0, 255); text-emphasis-color: rgb(14, 0, 255); \"\u003eend\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003en = 8;\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003em = 3;\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003enogo = 5;\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eavgmsim = simulategame(n,m,nogo,1e4)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eavgmsim =\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e6.3626\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThis approach would require too many simulations to be accurate to the required precision, but gives rough verification that the game takes 6.375 moves on average.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor n = 16, m = 5, and the forbidden squares are 4, 6, 10, 11, and 15, the transition matrix is:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 429.188px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 641.075px 214.587px; transform-origin: 641.081px 214.594px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eidx = [1;17;18;33;34;35;65;66;67;69;98;99;101;103;115;117;119;120;133;135;136;137;183;184;185;188;200;201;204;205;217;220;221;222;252;253;254;256];\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eTvals = [0.4;0.2;0.4;0.2;0.2;0.4;0.2;0.2;0.2;0.4;0.2;0.2;0.2;0.4;0.2;0.2;0.2;0.4;0.2;0.2;0.2;0.4;0.2;0.2;0.2;0.4;0.2;0.2;0.2;0.6;0.2;0.2;0.2;0.8;0.2;0.2;0.2;1];\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eT = zeros(16);\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eT(idx) = Tvals\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eT =\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e0.4000    0.2000    0.2000         0    0.2000         0         0         0         0         0         0         0         0         0         0         0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e0    0.4000    0.2000         0    0.2000         0    0.2000         0         0         0         0         0         0         0         0         0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e0         0    0.4000         0    0.2000         0    0.2000    0.2000         0         0         0         0         0         0         0         0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e0         0         0         0    0.4000         0    0.2000    0.2000    0.2000         0         0         0         0         0         0         0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e0         0         0         0         0         0    0.4000    0.2000    0.2000         0         0    0.2000         0         0         0         0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e0         0         0         0         0         0         0    0.4000    0.2000         0         0    0.2000    0.2000         0         0         0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e0         0         0         0         0         0         0         0    0.4000         0         0    0.2000    0.2000    0.2000         0         0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e0         0         0         0         0         0         0         0         0         0         0    0.4000    0.2000    0.2000         0    0.2000\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e0         0         0         0         0         0         0         0         0         0         0         0    0.6000    0.2000         0    0.2000\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e0         0         0         0         0         0         0         0         0         0         0         0         0    0.8000         0    0.2000\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e0         0         0         0         0         0         0         0         0         0         0         0         0         0         0    1.0000\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eComputing the average number of moves from the transition matrix and from simulation:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 122.625px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 641.075px 61.3125px; transform-origin: 641.081px 61.3125px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eavgm = avgmoves(T)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eavgm =\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e11.4015\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eavgmsim = simulategame(16,5,[4 6 10 11 15],1e4)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003eavgmsim =\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 641.075px 10.2125px; text-wrap-mode: nowrap; transform-origin: 641.081px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; unicode-bidi: normal; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e11.4266\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eAssumptions\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe input will be a valid transition matrix for a game with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003en\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u0026gt; \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003em\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e ≥ 2. The output should be a double-precision scalar that has the average number of moves, accurate to at least 0.001.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function mv = avgmoves(T)\r\nmv = 1 + (1-T(1,end)); % m=0 + m=1\r\nend","test_suite":"%% Test case #1: m = 3, n = 28, nogo = [3 7 13 18 21]\r\nidx = [1;29;30;85;86;88;114;116;117;144;145;146;201;202;230;232;260;261;262;288;289;290;291;317;318;319;320;375;376;404;406;407;434;435;436;462;463;464;465;520;521;523;549;551;552;607;608;636;638;666;667;694;695;696;723;724;725;726;752;753;754;755;781;782;783;784];\r\nTvals = [1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;2;1;1;1;3]/3;\r\nT = zeros(28);\r\nT(idx) = Tvals;\r\nmavg = avgmoves(T);\r\nassert( abs(mavg - 19.09765625) \u003c 0.001 )\r\n\r\n%% Test case #2: m = 3, n = 25, nogo = [8 17 18 22]\r\nidx = [26;51;52;76;77;78;102;103;104;105;128;129;130;131;154;155;156;157;206;207;232;234;259;260;284;285;286;310;311;312;336;337;338;339;362;363;364;365;388;389;390;391;466;469;494;495;519;520;521;570;571;573;596;598;599;623;624;625];\r\nTvals = [1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;2;1;1;1;2;1;1;1;1;1;1;1;1;1;1;1;1;2;1;1;3]/3;\r\nT = zeros(25);\r\nT(idx) = Tvals;\r\nmavg = avgmoves(T);\r\nassert( abs(mavg - 19.52023320) \u003c 0.001 )\r\n\r\n%% Test case #3: m = 4, n = 10, nogo = [4 5 6]\r\nidx = [1;11;12;21;22;23;63;67;77;78;87;88;89;97;98;99;100];\r\nTvals = [2;1;3;1;1;3;1;1;1;2;1;1;3;1;1;1;4]/4;\r\nT = zeros(10);\r\nT(idx) = Tvals;\r\nmavg = avgmoves(T);\r\nassert( abs(mavg - 12.00000000) \u003c 0.001 )\r\n\r\n%% Test case #4: m = 4, n = 26, nogo = [3 13]\r\nidx = [1;27;28;79;80;105;106;108;132;134;135;160;161;162;186;187;188;189;213;214;215;216;217;240;241;242;243;244;267;268;269;270;271;294;295;296;297;298;348;349;350;375;376;378;402;404;405;430;431;432;456;457;458;459;483;484;485;486;510;511;512;513;537;538;539;540;564;565;566;567;591;592;593;594;595;618;619;620;621;622;645;646;647;648;649;672;673;674;675;676];\r\nTvals = [1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;2;1;1;1;1;3;1;1;1;1;4]/4;\r\nT = zeros(26);\r\nT(idx) = Tvals;\r\nmavg = avgmoves(T);\r\nassert( abs(mavg - 13.69784774) \u003c 0.001 )\r\n\r\n%% Test case #5: m = 5, n = 12, nogo = [5 6 8 9 11]\r\nidx = [1;13;14;25;26;27;37;38;39;40;74;75;76;79;115;118;139;142;144];\r\nTvals = [2;1;2;1;1;3;1;1;1;4;1;1;1;3;1;4;1;1;5]/5;\r\nT = zeros(12);\r\nT(idx) = Tvals;\r\nmavg = avgmoves(T);\r\nassert( abs(mavg - 11.66666667) \u003c 0.001 )\r\n\r\n%% Test case #6: m = 5, n = 18, nogo = [5 6 9 10]\r\nidx = [1;19;20;37;38;39;55;56;57;58;110;111;112;115;129;130;133;134;187;188;205;206;209;224;227;228;245;246;247;248;263;264;265;266;267;281;282;283;284;285;286;300;301;302;303;304;305;319;320;321;322;323;324];\r\nTvals = [2;1;2;1;1;2;1;1;1;3;1;1;1;2;1;1;1;2;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;2;1;1;1;1;1;3;1;1;1;1;1;4;1;1;1;1;1;5]/5;\r\nT = zeros(18);\r\nT(idx) = Tvals;\r\nmavg = avgmoves(T);\r\nassert( abs(mavg - 12.28888889) \u003c 0.001 )\r\n\r\n%% Test case #7: m = 6, n = 8, nogo = []\r\nidx = [9;17;18;19;25;26;27;28;33;34;35;36;37;41;42;43;44;45;46;49;50;51;52;53;54;55;58;59;60;61;62;63;64];\r\nTvals = [1;1;1;1;1;1;1;2;1;1;1;1;3;1;1;1;1;1;4;1;1;1;1;1;1;5;1;1;1;1;1;1;6]/6;\r\nT = zeros(8);\r\nT(idx) = Tvals;\r\nmavg = avgmoves(T);\r\nassert( abs(mavg - 7.00000000) \u003c 0.001 )\r\n\r\n%% Test case #8: m = 6, n = 18, nogo = []\r\nidx = [19;37;38;55;56;57;73;74;75;76;91;92;93;94;95;109;110;111;112;113;114;128;129;130;131;132;133;147;148;149;150;151;152;166;167;168;169;170;171;185;186;187;188;189;190;204;205;206;207;208;209;223;224;225;226;227;228;229;242;243;244;245;246;247;248;261;262;263;264;265;266;267;280;281;282;283;284;285;286;299;300;301;302;303;304;305;318;319;320;321;322;323;324];\r\nTvals = [1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;2;1;1;1;1;1;1;3;1;1;1;1;1;1;4;1;1;1;1;1;1;5;1;1;1;1;1;1;6]/6;\r\nT = zeros(18);\r\nT(idx) = Tvals;\r\nmavg = avgmoves(T);\r\nassert( abs(mavg - 9.61298219) \u003c 0.001 )\r\n\r\n%% Test case #9: m = 7, n = 21, nogo = 3\r\nidx = [1;22;23;64;65;85;86;88;106;107;109;110;127;128;130;131;132;148;149;151;152;153;154;170;172;173;174;175;176;193;194;195;196;197;198;214;215;216;217;218;219;220;236;237;238;239;240;241;242;258;259;260;261;262;263;264;280;281;282;283;284;285;286;302;303;304;305;306;307;308;309;324;325;326;327;328;329;330;331;346;347;348;349;350;351;352;353;368;369;370;371;372;373;374;375;390;391;392;393;394;395;396;397;412;413;414;415;416;417;418;419;434;435;436;437;438;439;440;441];\r\nTvals = [1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;2;1;1;1;1;1;1;1;3;1;1;1;1;1;1;1;4;1;1;1;1;1;1;1;5;1;1;1;1;1;1;1;6;1;1;1;1;1;1;1;7]/7;\r\nT = zeros(21);\r\nT(idx) = Tvals;\r\nmavg = avgmoves(T);\r\nassert( abs(mavg - 10.83736048) \u003c 0.001 )\r\n\r\n%% Test case #10: m = 7, n = 9, nogo = [2 8]\r\nidx = [1;19;21;28;30;31;37;39;40;41;46;48;49;50;51;55;57;58;59;60;61;75;76;77;78;79;81];\r\nTvals = [2;1;2;1;1;3;1;1;1;4;1;1;1;1;5;1;1;1;1;1;6;1;1;1;1;1;7]/7;\r\nT = zeros(9);\r\nT(idx) = Tvals;\r\nmavg = avgmoves(T);\r\nassert( abs(mavg - 8.40000000) \u003c 0.001 )\r\n\r\n%% Test case #11: m = 8, n = 17, nogo = [4 6 9 16]\r\nidx = [1;18;19;35;36;37;69;70;71;73;103;104;105;107;109;120;121;122;124;126;127;155;156;158;160;161;163;173;175;177;178;180;181;192;194;195;197;198;199;209;211;212;214;215;216;217;228;229;231;232;233;234;235;245;246;248;249;250;251;252;253;282;283;284;285;286;287;289];\r\nTvals = [3;1;3;1;1;3;1;1;1;2;1;1;1;1;1;1;1;1;1;1;2;1;1;1;1;1;2;1;1;1;1;1;3;1;1;1;1;1;4;1;1;1;1;1;1;5;1;1;1;1;1;1;6;1;1;1;1;1;1;1;7;1;1;1;1;1;1;8]/8;\r\nT = zeros(17);\r\nT(idx) = Tvals;\r\nmavg = avgmoves(T);\r\nassert( abs(mavg - 11.58400000) \u003c 0.001 )\r\n\r\n%% Test case #12: m = 8, n = 9, nogo = [3 5 7]\r\nidx = [1;10;11;28;29;31;46;47;49;51;64;65;67;69;71;73;74;76;78;80;81];\r\nTvals = [3;1;4;1;1;5;1;1;1;6;1;1;1;1;7;1;1;1;1;1;8]/8;\r\nT = zeros(9);\r\nT(idx) = Tvals;\r\nmavg = avgmoves(T);\r\nassert( abs(mavg - 8.00000000) \u003c 0.001 )\r\n\r\n%% Test case #13: m = 9, n = 19, nogo = [8 9 13]\r\nidx = [1;20;21;39;40;41;58;59;60;61;77;78;79;80;81;96;97;98;99;100;101;115;116;117;118;119;120;121;172;173;174;175;176;177;178;181;192;193;194;195;196;197;200;201;212;213;214;215;216;219;220;221;252;253;254;257;258;259;261;272;273;276;277;278;280;281;292;295;296;297;299;300;301;314;315;316;318;319;320;321;333;334;335;337;338;339;340;341;352;353;354;356;357;358;359;360;361];\r\nTvals = [2;1;2;1;1;2;1;1;1;3;1;1;1;1;3;1;1;1;1;1;3;1;1;1;1;1;1;3;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;2;1;1;1;1;1;1;1;3;1;1;1;1;1;1;4;1;1;1;1;1;1;5;1;1;1;1;1;1;6;1;1;1;1;1;1;7;1;1;1;1;1;1;1;8;1;1;1;1;1;1;1;1;9]/9;\r\nT = zeros(19);\r\nT(idx) = Tvals;\r\nmavg = avgmoves(T);\r\nassert( abs(mavg - 12.11111111) \u003c 0.001 )\r\n\r\n%% Test case #14: m = 9, n = 29, nogo = 13\r\nidx = [30;59;60;88;89;90;91;117;118;119;120;121;146;147;148;149;150;151;175;176;177;178;179;180;181;204;205;206;207;208;209;210;211;233;234;235;236;237;238;239;240;241;262;263;264;265;266;267;268;269;270;271;292;293;294;295;296;297;298;299;300;301;322;323;324;325;326;327;328;329;330;331;382;383;384;385;386;387;388;389;412;413;414;415;416;417;418;420;442;443;444;445;446;447;449;450;472;473;474;475;476;478;479;480;502;503;504;505;507;508;509;510;532;533;534;536;537;538;539;540;562;563;565;566;567;568;569;570;592;594;595;596;597;598;599;600;601;623;624;625;626;627;628;629;630;631;652;653;654;655;656;657;658;659;660;661;682;683;684;685;686;687;688;689;690;691;712;713;714;715;716;717;718;719;720;721;742;743;744;745;746;747;748;749;750;751;772;773;774;775;776;777;778;779;780;781;802;803;804;805;806;807;808;809;810;811;832;833;834;835;836;837;838;839;840;841];\r\nTvals = [1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;2;1;1;1;1;1;1;1;1;1;3;1;1;1;1;1;1;1;1;1;4;1;1;1;1;1;1;1;1;1;5;1;1;1;1;1;1;1;1;1;6;1;1;1;1;1;1;1;1;1;7;1;1;1;1;1;1;1;1;1;8;1;1;1;1;1;1;1;1;1;9]/9;\r\nT = zeros(29);\r\nT(idx) = Tvals;\r\nmavg = avgmoves(T);\r\nassert( abs(mavg - 13.56167777) \u003c 0.001 )\r\n\r\n%% Test case #15: m = 10, n = 15, nogo = [7 10]\r\nidx = [1;16;17;31;32;33;46;47;48;49;61;62;63;64;65;76;77;78;79;80;81;106;107;108;109;110;111;113;121;122;123;124;125;126;128;129;151;152;153;154;155;156;158;159;161;167;168;169;170;171;173;174;176;177;183;184;185;186;188;189;191;192;193;199;200;201;203;204;206;207;208;209;215;216;218;219;221;222;223;224;225];\r\nTvals = [2;1;2;1;1;2;1;1;1;2;1;1;1;1;2;1;1;1;1;1;3;1;1;1;1;1;1;4;1;1;1;1;1;1;1;5;1;1;1;1;1;1;1;1;6;1;1;1;1;1;1;1;1;7;1;1;1;1;1;1;1;1;8;1;1;1;1;1;1;1;1;9;1;1;1;1;1;1;1;1;10]/10;\r\nT = zeros(15);\r\nT(idx) = Tvals;\r\nmavg = avgmoves(T);\r\nassert( abs(mavg - 11.77978516) \u003c 0.001 )\r\n\r\n%% Test case #16: m = 10, n = 23, nogo = 14\r\nidx = [24;47;48;70;71;72;73;93;94;95;96;97;116;117;118;119;120;121;139;140;141;142;143;144;145;162;163;164;165;166;167;168;169;185;186;187;188;189;190;191;192;193;208;209;210;211;212;213;214;215;216;217;231;232;233;234;235;236;237;238;239;240;241;255;256;257;258;259;260;261;262;263;264;265;279;280;281;282;283;284;285;286;287;288;289;327;328;329;330;331;332;333;334;335;337;351;352;353;354;355;356;357;358;360;361;375;376;377;378;379;380;381;383;384;385;399;400;401;402;403;404;406;407;408;409;423;424;425;426;427;429;430;431;432;433;447;448;449;450;452;453;454;455;456;457;471;472;473;475;476;477;478;479;480;481;495;496;498;499;500;501;502;503;504;505;519;521;522;523;524;525;526;527;528;529];\r\nTvals = [1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;2;1;1;1;1;1;1;1;1;1;3;1;1;1;1;1;1;1;1;1;4;1;1;1;1;1;1;1;1;1;5;1;1;1;1;1;1;1;1;1;6;1;1;1;1;1;1;1;1;1;7;1;1;1;1;1;1;1;1;1;8;1;1;1;1;1;1;1;1;1;9;1;1;1;1;1;1;1;1;1;10]/10;\r\nT = zeros(23);\r\nT(idx) = Tvals;\r\nmavg = avgmoves(T);\r\nassert( abs(mavg - 13.01211039) \u003c 0.001 )\r\n\r\n%% Test case #17: m = 11, n = 13, nogo = [6 7 11]\r\nidx = [1;14;15;27;28;29;40;41;42;43;53;54;55;56;57;92;93;94;95;96;99;105;106;107;108;109;112;113;118;119;120;121;122;125;126;127;144;145;146;147;148;151;152;153;155;158;159;160;161;164;165;166;168;169];\r\nTvals = [3;1;3;1;1;4;1;1;1;5;1;1;1;1;6;1;1;1;1;1;7;1;1;1;1;1;1;8;1;1;1;1;1;1;1;9;1;1;1;1;1;1;1;1;10;1;1;1;1;1;1;1;1;11]/11;\r\nT = zeros(13);\r\nT(idx) = Tvals;\r\nmavg = avgmoves(T);\r\nassert( abs(mavg - 12.37500000) \u003c 0.001 )\r\n\r\n%% Test case #18: m = 11, n = 18, nogo = []\r\nidx = [19;37;38;55;56;57;73;74;75;76;91;92;93;94;95;109;110;111;112;113;114;127;128;129;130;131;132;133;134;145;146;147;148;149;150;151;152;153;163;164;165;166;167;168;169;170;171;172;181;182;183;184;185;186;187;188;189;190;191;199;200;201;202;203;204;205;206;207;208;209;210;218;219;220;221;222;223;224;225;226;227;228;229;237;238;239;240;241;242;243;244;245;246;247;248;256;257;258;259;260;261;262;263;264;265;266;267;275;276;277;278;279;280;281;282;283;284;285;286;294;295;296;297;298;299;300;301;302;303;304;305;313;314;315;316;317;318;319;320;321;322;323;324];\r\nTvals = [1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;2;1;1;1;1;1;1;1;1;1;3;1;1;1;1;1;1;1;1;1;1;4;1;1;1;1;1;1;1;1;1;1;1;5;1;1;1;1;1;1;1;1;1;1;1;6;1;1;1;1;1;1;1;1;1;1;1;7;1;1;1;1;1;1;1;1;1;1;1;8;1;1;1;1;1;1;1;1;1;1;1;9;1;1;1;1;1;1;1;1;1;1;1;10;1;1;1;1;1;1;1;1;1;1;1;11]/11;\r\nT = zeros(18);\r\nT(idx) = Tvals;\r\nmavg = avgmoves(T);\r\nassert( abs(mavg - 12.54505095) \u003c 0.001 )\r\n\r\n%% Test case #19: m = 12, n = 19, nogo = [2 10 18]\r\nidx = [1;39;41;58;60;61;77;79;80;81;96;98;99;100;101;115;117;118;119;120;121;134;136;137;138;139;140;141;153;155;156;157;158;159;160;161;191;193;194;195;196;197;198;199;201;210;212;213;214;215;216;217;218;220;221;229;231;232;233;234;235;236;237;239;240;241;250;251;252;253;254;255;256;258;259;260;261;269;270;271;272;273;274;275;277;278;279;280;281;289;290;291;292;293;294;296;297;298;299;300;301;309;310;311;312;313;315;316;317;318;319;320;321;349;350;351;353;354;355;356;357;358;359;361];\r\nTvals = [2;1;1;1;1;1;1;1;1;1;1;1;1;1;2;1;1;1;1;1;2;1;1;1;1;1;1;3;1;1;1;1;1;1;1;4;1;1;1;1;1;1;1;1;5;1;1;1;1;1;1;1;1;1;6;1;1;1;1;1;1;1;1;1;1;7;1;1;1;1;1;1;1;1;1;1;8;1;1;1;1;1;1;1;1;1;1;1;9;1;1;1;1;1;1;1;1;1;1;1;10;1;1;1;1;1;1;1;1;1;1;1;11;1;1;1;1;1;1;1;1;1;1;12]/12;\r\nT = zeros(19);\r\nT(idx) = Tvals;\r\nmavg = avgmoves(T);\r\nassert( abs(mavg - 13.71371901) \u003c 0.001 )\r\n\r\n%% Test case #20: m = 12, n = 23, nogo = [6 7 11 14]\r\nidx = [1;24;25;47;48;49;70;71;72;73;93;94;95;96;97;162;163;164;165;166;169;185;186;187;188;189;192;193;208;209;210;211;212;215;216;217;254;255;256;257;258;261;262;263;265;277;278;279;280;281;284;285;286;288;289;325;326;327;330;331;332;334;335;337;349;350;353;354;355;357;358;360;361;373;376;377;378;380;381;383;384;385;399;400;401;403;404;406;407;408;409;422;423;424;426;427;429;430;431;432;433;445;446;447;449;450;452;453;454;455;456;457;469;470;472;473;475;476;477;478;479;480;481;493;495;496;498;499;500;501;502;503;504;505;518;519;521;522;523;524;525;526;527;528;529];\r\nTvals = [3;1;4;1;1;4;1;1;1;4;1;1;1;1;4;1;1;1;1;1;2;1;1;1;1;1;1;2;1;1;1;1;1;1;1;2;1;1;1;1;1;1;1;1;2;1;1;1;1;1;1;1;1;1;3;1;1;1;1;1;1;1;1;4;1;1;1;1;1;1;1;1;5;1;1;1;1;1;1;1;1;6;1;1;1;1;1;1;1;1;7;1;1;1;1;1;1;1;1;1;8;1;1;1;1;1;1;1;1;1;1;9;1;1;1;1;1;1;1;1;1;1;10;1;1;1;1;1;1;1;1;1;1;11;1;1;1;1;1;1;1;1;1;1;12]/12;\r\nT = zeros(23);\r\nT(idx) = Tvals;\r\nmavg = avgmoves(T);\r\nassert( abs(mavg - 14.84267285) \u003c 0.001 )","published":true,"deleted":false,"likes_count":3,"comments_count":0,"created_by":287,"edited_by":287,"edited_at":"2025-11-10T02:21:25.000Z","deleted_by":null,"deleted_at":null,"solvers_count":82,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2025-11-04T20:41:43.000Z","updated_at":"2026-02-13T10:03:15.000Z","published_at":"2025-11-10T02:21:25.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eBored of tedious court assemblies, King \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eNeduchadneddar\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e the Procrastinator has found a cunning way to keep his advisors, councilors, viziers, and other courtly lackeys busy: he plays a simple and pointless board game with them.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ePlayers move their pawn along a path of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e squares, starting at square 1, with the goal of being the first to reach square \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e. On each turn, they roll an \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e-sided die to determine how many squares to move. However, if their move would take them beyond the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eth\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e square, they stay put. Also, players cannot land on any of the numbers that King \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eNeduchadneddar\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e has decided are the unlucky numbers of the day. If their move would land them on any of these squares, they stay put instead.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eTo keep his courtiers on their toes (so they have less time to plot his downfall), the king changes the number of squares (\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e), the number of sides of the die (\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e), and the list of unlucky forbidden squares each day.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eNeeding to know how much time he will waste with each game, the king in his wisdom (and/or insanity) has appointed you Chief Royal Mage of Matrix Computations and tasked you with determining the average number of moves it will take a player to complete a game according to today’s setup.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eYou can use the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003etransition matrix\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e for the game to do this. The transition matrix \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eT\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e is an \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e-by-\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e matrix where T(\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ej\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e,\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ek\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e) is the probability of moving from square \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ej\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e to square \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ek\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e in one move. Then \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"21\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"51\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e gives the probability of moving from from square \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ej\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e to square \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ek\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e in \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e moves (where \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"20\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"20\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is a matrix power; that is, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"18\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"84\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId3\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, using standard matrix multiplication).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"21\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"52\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId4\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the probability of moving from square 1 to square \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e in \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e moves. That is, it is the probability of completing the game in \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e moves. However, once you reach square \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e you stay there forever. (In practice, the game would end at this point.) Therefore, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"21\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"52\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId4\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e increases monotonically towards 1 as \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e increases, and can be interpreted as the cumulative probability of completing the game in \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e moves. Or, in other words, the probability of completing the game in \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e moves or fewer.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThis is convenient because there's a result in probability theory that allows you to calculate the expected value of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e (that is, the average number of moves to complete the game) from the cumulative probability. Skipping the math details, the punchline is:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"49\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"136\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId5\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAn infinite sum may not seem helpful, but remember that \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"21\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"52\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId4\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e increases monotonically towards 1, which means the terms in the sum go to zero. Therefore, in practice, you can find the average number of moves to complete the game by summing \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"21\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"76\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId6\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e until the terms are too small to make any practical difference.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven a transition matrix \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eT\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, return the average number of moves \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"19\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"14\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId7\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, accurate to 3 decimal places.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExample\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWith \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e = 8 and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e = 3, and square 5 forbidden, the transition matrix is:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[T = [0      1/3    1/3    1/3    0      0      0      0       \\n    0      1/3    1/3    1/3    0      0      0      0       \\n    0      0      1/3    1/3    0      1/3    0      0       \\n    0      0      0      1/3    0      1/3    1/3    0       \\n    0      0      0      0      0      0      0      0       \\n    0      0      0      0      0      1/3    1/3    1/3     \\n    0      0      0      0      0      0      2/3    1/3     \\n    0      0      0      0      0      0      0      1];\\n\\navgm = avgmoves(T)\\navgm =\\n6.3750]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAn inexact way to verify the result is to directly simulate playing the game many times:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[function avgm = simulategame(n,m,nogo,nexp)\\nmv = zeros(nexp,1);\\nfor g = 1:nexp\\n    % Start game at 1\\n    s = 1;\\n    % Go until we reach the nth square\\n    while (s \u003c n)\\n        % Roll die and see where we would land\\n        snext = s + randi(m);\\n        if (snext \u003c= n) \u0026\u0026 ~any(snext == nogo)\\n            % Move only if valid\\n            s = snext;\\n        end\\n        % Increment number of moves for this game\\n        mv(g) = mv(g) + 1;\\n    end\\n    % Go to the next game\\nend\\navgm = mean(mv);\\nend\\n\\nn = 8;\\nm = 3;\\nnogo = 5;\\navgmsim = simulategame(n,m,nogo,1e4)\\navgmsim =\\n6.3626]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThis approach would require too many simulations to be accurate to the required precision, but gives rough verification that the game takes 6.375 moves on average.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor n = 16, m = 5, and the forbidden squares are 4, 6, 10, 11, and 15, the transition matrix is:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[idx = [1;17;18;33;34;35;65;66;67;69;98;99;101;103;115;117;119;120;133;135;136;137;183;184;185;188;200;201;204;205;217;220;221;222;252;253;254;256];\\nTvals = [0.4;0.2;0.4;0.2;0.2;0.4;0.2;0.2;0.2;0.4;0.2;0.2;0.2;0.4;0.2;0.2;0.2;0.4;0.2;0.2;0.2;0.4;0.2;0.2;0.2;0.4;0.2;0.2;0.2;0.6;0.2;0.2;0.2;0.8;0.2;0.2;0.2;1];\\nT = zeros(16);\\nT(idx) = Tvals\\nT =\\n0.4000    0.2000    0.2000         0    0.2000         0         0         0         0         0         0         0         0         0         0         0\\n0    0.4000    0.2000         0    0.2000         0    0.2000         0         0         0         0         0         0         0         0         0\\n0         0    0.4000         0    0.2000         0    0.2000    0.2000         0         0         0         0         0         0         0         0\\n0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0\\n0         0         0         0    0.4000         0    0.2000    0.2000    0.2000         0         0         0         0         0         0         0\\n0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0\\n0         0         0         0         0         0    0.4000    0.2000    0.2000         0         0    0.2000         0         0         0         0\\n0         0         0         0         0         0         0    0.4000    0.2000         0         0    0.2000    0.2000         0         0         0\\n0         0         0         0         0         0         0         0    0.4000         0         0    0.2000    0.2000    0.2000         0         0\\n0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0\\n0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0\\n0         0         0         0         0         0         0         0         0         0         0    0.4000    0.2000    0.2000         0    0.2000\\n0         0         0         0         0         0         0         0         0         0         0         0    0.6000    0.2000         0    0.2000\\n0         0         0         0         0         0         0         0         0         0         0         0         0    0.8000         0    0.2000\\n0         0         0         0         0         0         0         0         0         0         0         0         0         0         0         0\\n0         0         0         0         0         0         0         0         0         0         0         0         0         0         0    1.0000]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eComputing the average number of moves from the transition matrix and from simulation:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[avgm = avgmoves(T)\\navgm =\\n11.4015\\navgmsim = simulategame(16,5,[4 6 10 11 15],1e4)\\navgmsim =\\n11.4266]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eAssumptions\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe input will be a valid transition matrix for a game with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u0026gt; \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e ≥ 2. The output should be a double-precision scalar that has the average number of moves, accurate to at least 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