Rich writes: << I've shown N^3 TicTacToe is drawn for N >= 7; N^4 is drawn for N >= 11; N^5 is drawn for N >= 14; ... N^D is drawn for N >= 3 D - (D mod 2) (and D>=3).

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This is fascinating, and (if it hasn't already been published) sounds well worthy of writing up as an article for the Math. Intelligencer or the American Mathematical Monthly. Rich, do you know for which N the first player is known to have a winning strategy? * * * I'll mention two torus games, the first of which I've already alluded to: 1. n^D toral-tic-tac-toe (T4). This just assumes the usual wraparound when a line gets to the edge of the board. The winning "lines" would actually be certain "finite circles" in the torus, since there'd be no distinguished start or end. E.g., in the 4^3 case, there'd be the same 3x16 = 48 axis-aligned wins as in cubical tic-tac-toe, but a total of 6x16 = 96 wins parallel to a face diagonal (compared to 24 in the cubical case) and a total of 4x16 = 64 wins parallel to a main diagonal (compared to 4). So the total number of winning "lines" is 13x16 = 208 (compared to 76). 2. n^D affine toral tic-tac-toe (AT4). This is easiest to describe mathematically: A win is any "affine line" (which again is a finite "circle"). Think of the board as the abelian group G = (Z_n)^D where Z_n = integers mod n. A win is any coset of any subgroup H isomorphic to Z_n, or in other words a subset L of the form g + H where g is any element of G. For the 4^3 game, each subgroup isomorphic to Z_4 has 64/4 = 16 cosets, so the number of winning "lines" is 16 x (the number of subgroups of (Z_4)^3 that are isomorphic to Z_4). If I'm not mistaken there are 28 distinct subgroups isomorphic to Z_4, and so 16x28 = 448 different wins. * * * Can the first player always win 4^3 AT4 ??? How about 4^4 AT4 ??? --Dan