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 draw on an infinite board means that the game will go on forever with perfect play. k ≥ 9 is a draw: when k = 9 and the board is infinite, the second player can draw via a "pairing strategy".Likewise, if ( m 0, n 0, k 0) is a win, then ( m 0, n 0, k) with k ≤ k 0 is a win, and ( m, n, k 0) with m ≥ m 0 and n ≥ n 0 is a win. If a particular ( m 0, n 0, k 0) is a draw, then ( m 0, n 0, k) with k ≥ k 0 is a draw, and ( m, n, k 0) with m ≤ m 0 and n ≤ n 0 is a draw.The following statements refer to the first player in the weak game, assuming that both players use an optimal strategy. Note that proofs of draws using pairing strategies also prove a draw for the weak version and thus for all smaller versions. If weak ( m, n, k) is a draw, then decreasing m or n, or increasing k will also result in a drawn game.Ĭonversely, if weak or normal ( m, n, k) is a win, then any larger weak ( m, n, k) is a win. Also, it does not actually give a strategy for the first player.Īpplying results to different board sizes Ī 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. This argument tells nothing about whether a particular game is a draw or a win for the first player. The contradiction implies that the original assumption is false, and the second player cannot have a winning strategy. Since an extra stone cannot hurt them, this is a winning strategy for the first player. If this happens, though, they can again play an arbitrary move and continue as before with the second player's winning strategy. They can do this as long as the strategy doesn't call for placing a stone on the 'arbitrary' square that is already occupied. After that, the player pretends that they are the second player and adopts the second player's winning strategy. The first player makes an arbitrary move, to begin with. The strategy stealing argument assumes that the second player has a winning strategy and demonstrates a winning strategy for the first player. This is because an extra stone given to either player in any position can only improve that player's chances.
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