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Tic-tac-toe is a game traditionally played in a 2-dimensional 3 by 3 grid, where each cell of the grid can either be an "X", an "O", or it can be empty, and the number of cells empty is the number of moves played subtracted from 9 (so a game can never exceed 9 moves). A game where at least 5 moves has been played can have a winner, but it also can continue going, if in no row, column, or diagonal has 3 of the same symbol in it. 3 empty cells in a row, column, or diagonal does not count, as those have no symbol and thus no player attached to them. Incidentally, due to the alternating turns of the two players, the symmetry of the game board, and it's 3-by-3-ness, tie games are possible, where, after 9 turns, there are no rows, columns, or diagonals where there are 3 matching symbols in them. Is it possible, in higher dimensions, to construct a board with rows, columns, and their higher dimensional counterparts of length 3, such that the game always has a winner?

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marked as duplicate by Xander Henderson, JonMark Perry, ml0105, Rohan, Claude Leibovici Dec 31 '17 at 10:18

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    $\begingroup$ Your description of the two-dimensional game is detailed, but you largely hint at what is meant by the "higher dimensional counterparts of length 3". Rows and columns seem inadequate to "construct a board" of higher dimension. Compare this previous Question as a model for describing similar games. $\endgroup$ – hardmath Dec 31 '17 at 2:47
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    $\begingroup$ The Hales-Jewett therorem en.wikipedia.org/wiki/Hales%E2%80%93Jewett_theorem states that a suitable large enough boards can never be drawn. $\endgroup$ – Lord Shark the Unknown Dec 31 '17 at 5:16
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For the 3 by 3 board in 3 dimensions, the first player will always win with optimal play. Generalizations of $n$ by $n$ boards in $d$ dimensions is explored in this very nice video by PBS Infinite Series.

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  • $\begingroup$ In my experience, the only way to make higher-dimension tic-tac-toe fun is to forbid playing in the center space, except when forced or as a finishing move. It also seems to be the case that the best games occur when the number of players is one less than the number of dimensions. $\endgroup$ – G Tony Jacobs Dec 31 '17 at 2:51
  • $\begingroup$ Thank you! Is there anywhere I can view a proof of this conjecture? It isn't stated in the video. $\endgroup$ – Matthew Sylvester Dec 31 '17 at 3:04

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