A n<sup>d</sup> game (or n<sup>k</sup> game) is a generalization of the combinatorial game tic-tac-toe to higher dimensions. It is a game played on a n<sup>d</sup> hypercube with 2 players. If one player creates a line of length n of their symbol (X or O) they win the game. However, if all n<sup>d</sup> spaces are filled then the game is a draw. Tic-tac-toe is the game where n equals 3 and d equals 2 (3, 2). Qubic is the game. The or games are trivially won by the first player as there is only one space ( and ). A game with and cannot be won if both players are playing well as an opponent's piece will block the one-dimensional line.
An n<sup>d</sup> game is a symmetric combinatorial game.
There are a total of winning lines in a n<sup>d</sup> game.
For any width n, at some dimension d (thanks to the Hales-Jewett theorem), there will always be a winning strategy for player X. There will never be a winning strategy for player O because of the Strategy-stealing argument since an n<sup>d</sup> game is symmetric.