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This argument tells nothing about whether a particular game is a draw or a win for the first player. Also, it does not actually give a strategy for the first player.
Applying results to different board sizes[edit]
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A useful notion is a 'weak (m,n,k) game', where k-in-a-row by the second player does not end the game with a second player win.
https://renobrown600.weebly.com/code-orange-forever-album-free-download-torrent.html. If weak (m,n,k) is a draw, then decreasing m or n, or increasing k will also result in a drawn game.
Conversely, if weak or normal (m,n,k) is a win, then any larger weak (m,n,k) is a win.
Note that proofs of draws using pairing strategies also prove a draw for the weak version and thus for all smaller versions.
General results[edit]
The following statements refer to the first player, assuming that both players use an optimal strategy.
- k ≥ 9 is a draw: when k = 9 and the board is infinite, the second player can draw via a 'pairing strategy'. A draw on an infinite board means that the game will go on forever with perfect play. A pairing strategy involves dividing all the squares of the board into pairs in such a way that by always playing on the pair of the first player's square, the second player is ensured that the first player cannot get k in a line. A pairing strategy on an infinite board can be applied to any finite board as well - if the strategy calls for making a move outside the board, then the second player makes an arbitrary move inside the board.[3]
- k ≥ 8 is a draw on an infinite board. It is not clear if this strategy applies to any finite board sizes.[2] It is not known if the second player can force a draw when k is 6 or 7 on an infinite board.
- k ≥ 3 and either k > m or k > n is a draw, also by a pairing strategy in the dimension not smaller than k (or trivially impossible to win if both are smaller)[3]
Specific results[edit]
- k = 1 and k = 2 are trivial wins, except for (1,1,2) and (2,1,2)
- k = 3 is a draw for (3,3,3) (see Tic-tac-toe) or if m < 3 or n < 3. It is a win otherwise.
- (m,4,4) is a win for m ≥ 30 (Lustenberger, 1967) and a draw for m ≤ 8.[1]
- (5,5,4) is a draw.
- (6,5,4) is a win.
- (6,6,5) is a draw.
- Computer search by Wei-Yuan Hsu and Chu-Ling Ko has shown that both (7,7,5) and (8,8,5) are draws.[4]
- Computer search by L. Victor Allis has shown that (15,15,5) is a win, even with one of the restrictive rules of Gomoku.
- (9,6,6) and (7,7,6) are both draws via pairings.
Multidimensional variant[edit]
It is possible to consider variants played on a multidimensional instead of a bidimensional board.
For the case of k-in-a-row where the board is an n-dimensional hypercube with all edges with length k, Hales and Jewett proved[5] that the game is a draw if k is odd and
- k ≥ 3n - 1
or if k is even and
- k ≥ 2n+1 - 2.
They conjecture that the game is a draw also when the number of cells is at least twice the number of lines, which happens if and only if
P N B Bank
- 2 kn ≥ (k + 2)n.
See also[edit]
References[edit]
- ^ abJ. W. H. M. Uiterwijk and H. J van der Herik, The advantage of the initiative, Information Sciences 122 (1) (2000) 43-58.
- ^ abJaap van den Herik, Jos W.H.M. Uiterwijk, Jack van Rijswijck (2002). 'Games solved: Now and in the future'. Artificial Intelligence.
- ^ abWei Ji Ma. 'Generalizations of tic-tac-toe'
- ^Hsu, W. Y., Ko, C. L., Hsueh, C. H., & Wu, I. 'Solving 7, 7, 5-game and 8, 8, 5-game'. ICGA Journal(2018).
- ^Elwyn R. Berlekamp, John Horton Conway, Richard K. Guy. 'Winning ways for your mathematical plays, Volume 3', A K Peters (2003)
External links[edit]
- W.J. Ma, Generalizations of tic-tac-toe, [1].
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