n^d game

A n^d game (or n^k game) is a generalization of the 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]

There are a total of ((n + 2)^d − n^d)/2 winning lines in a n^d game.[2][6]

See also

• Tic-tac-toe
• Treblecross

References

1. "Mathllaneous" (PDF). Retrieved December 16, 2016.
2. Beck, József (2008-03-20). Combinatorial Games: Tic-Tac-Toe Theory. Cambridge University Press. ISBN 9780521461009.
3. Tichy, Robert F.; Schlickewei, Hans Peter; Schmidt, Klaus D. (2008-07-10). Diophantine Approximation: Festschrift for Wolfgang Schmidt. Springer. ISBN 9783211742808.
4. Golomb, Solomon; Hales, Alfred. "Hypercube Tic-Tac-Toe" (PDF). Archived from the original (PDF) on April 29, 2016. Retrieved December 16, 2016.
5. Shih, Davis. "A Scientific Study: k-dimensional Tic-Tac-Toe" (PDF). Retrieved December 16, 2016.
6. Epstein, Richard A. (2012-12-28). The Theory of Gambling and Statistical Logic. Academic Press. ISBN 9780123978707.