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-by-n board, the winner being the player who first gets k stones of their own color in a row, horizontally, vertically, or diagonally.

They 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, they can again play an arbitrary move and continue as before with the second player's winning strategy.

The contradiction implies that the original assumption is false, and the second player cannot have a winning strategy.

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 or if k is even and 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