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M,n,k-game


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. m,n,k-game is also called a k-in-a-row game on m×n board.

m,n,k-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.

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, the player pretends that he is the second player and adopts the second player's winning strategy. He 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 can again play an arbitrary move and continue as before with the second player's winning strategy. Since an extra stone cannot hurt him, this is a winning strategy for the first player. The contradiction implies that the original assumption is false, and the second player cannot have a winning strategy.

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.

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.


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