n^d game

A n^d game (or n^k game) is a generalization of the combinatorial game tic-tac-toe to higher dimensions.^[1][2][3] It is a game played on a n^d hypercube with 2 players.^[1][2][4][5] If one player creates a line of length n of their symbol (X or O) they win the game. However, if all n^d spaces are filled then the game is a draw.^[4] Tic-tac-toe is the game where n equals 3 and d equals 2 (3, 2).^[4] Qubic is the (4, 3) game.^[4] The (n > 0, 0) or (1, 1) games are trivially won by the first player as there is only one space (n⁰ = 1 and 1¹ = 1). A game with d = 1 and n > 1 cannot be won if both players are playing well as an opponent's piece will block the one-dimensional line.^[5]

An n^d game is a symmetric combinatorial game.

There are a total of ${\displaystyle {\frac {\left(n+2\right)^{d}-n^{d}}{2}}}$ winning lines in a n^d game.^[2][6]

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^d game is symmetric.

• Treblecross – Degenerate tic-tac toe variant

• Higher-Dimensional Tic-Tac-Toe from the PBS Infinite Series on YouTube

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