A nd game (or nk game) is a generalization of the combinatorial game tic-tac-toe to higher dimensions.[1] [2] [3] It is a game played on a nd hypercube with 2 players.[4] [5] If one player creates a line of length n of their symbol (X or O) they win the game. However, if all nd 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 nd game is a symmetric combinatorial game.
There are a total of
| \left(n+2\right)d-nd | |
| 2 |
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 nd game is symmetric.