Games - M,n,k-games

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	Games - M,n,k-games

An m,n,k-game is an abstract board game in which two players take turns in placing a stone of their color on an m×n board, the winner being the player who first gets k stones of their own color in a row, horizontally, vertically, or diagonally. Thus, tic-tac-toe is the 3,3,3-game and free-style gomoku is the 19,19,5-game.

Apart from gomoku, mnk-games are mainly of mathematical interest. One seeks to find the game-theoretic value, which is the result of the game with perfect play. This is known as solving the game.

The second player cannot have a winning strategy

A standard strategy stealing argument from combinatorial game theory shows that in no m,n,k-game can there be a strategy that assures that the second player will win (a second-player winning strategy). This is because an extra stone given to either player in any position can only improve that player's chances. The strategy stealing argument assumes that the second player has a winning strategy and demonstrates a winning strategy for the first player. The first player makes an arbitrary move to begin with. After that, he or she pretends that he or she is the second player and adopts the second player's winning strategy. He or she can do this as long as the strategy doesn't call for placing a stone on the 'arbitrary' square that is already occupied. If this happens, though, he or she can again play an arbitrary move and continue as before with the second player's winning strategy. Since an extra stone can not hurt him or her, this is a winning strategy for the first player. The contradiction implies that the original assumption is false, and the second player can not have a winning strategy.

This argument tells us 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.

General results

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.
k ≥ 8 is a draw: on an infinite board. It is not clear if this strategy applies to any finite board sizes. It is not known if the second player can force a draw when k is 6 or 7 on an infinite board.
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) and a win otherwise if m ≥ 3 and n ≥ 3. If m < 3 or n < 3, the game is a draw.

Specific results

(m,4,4) is a win for m = 30 (Lustenberger, 1967) and a draw for m = 8. In 2003, it was shown to be a win for m = 9 (Sobotovych, see external link W.J. Ma).
(5,5,4) is a draw.
(6,5,4) is a win.
(6,6,5) is a draw.
Computer search by L. Victor Allis has shown that (15,15,5) is a win, even with one of the restrictive rules of Go-Moku.

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