nd game
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.[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 nd 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]
Game theory
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Unsolved problem in mathematics: Given a width of tic-tac-toe board, what is the smallest dimension such that X is guaranteed a winning strategy? (more unsolved problems in mathematics)
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An nd game is a symmetric combinatorial game.
There are a total of [math]\displaystyle{ \frac{\left(n+2\right)^d-n^d}{2} }[/math] winning lines in a nd game.[2][6]
For any width n, at some dimension k (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.
See also
References
- ↑ 1.0 1.1 "Mathllaneous". http://www.austms.org.au/Gazette/2005/Jul05/mathellaneous.pdf.
- ↑ 2.0 2.1 2.2 Beck, József (20 March 2008) (in en). Combinatorial Games: Tic-Tac-Toe Theory. Cambridge University Press. ISBN 9780521461009.
- ↑ Tichy, Robert F.; Schlickewei, Hans Peter; Schmidt, Klaus D. (10 July 2008) (in en). Diophantine Approximation: Festschrift for Wolfgang Schmidt. Springer. ISBN 9783211742808. https://books.google.com/books?id=7MeJRUdAwZUC&pg=PA46.
- ↑ 4.0 4.1 4.2 4.3 Golomb, Solomon; Hales, Alfred. "Hypercube Tic-Tac-Toe". http://library.msri.org/books/Book42/files/golomb.pdf.
- ↑ 5.0 5.1 Shih, Davis. "A Scientific Study: k-dimensional Tic-Tac-Toe". https://www.math.ucdavis.edu/~linear/student_creations/TicTacToe.pdf.
- ↑ Epstein, Richard A. (28 December 2012) (in en). The Theory of Gambling and Statistical Logic. Academic Press. ISBN 9780123978707. https://books.google.com/books?id=g5YWIpHTTW8C&dq=nd+tic+tac+toe+winning+lines&pg=PA341.
External links
- Higher-Dimensional Tic-Tac-Toe from the PBS Infinite Series on YouTube
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