Wikipedia:Reference desk/Archives/Mathematics/2023 July 4

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July 4

Harary's generalized tic-tac-toe for the pentominos and the hexominos

I proved that the Harary's generalized tic-tac-toe for the F, I, P, T, V, W, X, Z pentominos, the first player cannot win, but how about the L, S, U, Y pentominos? Can the first player win for these four pentominos? If so, what are the smallest size square board on which the first player can win and the smallest number of moves in which the first player can force a win, for these four pentominos?

Besides, I think the Harary's generalized tic-tac-toe for any of the 35 hexominos, the first player cannot win, can someone prove or disprove it? 118.170.49.79 (talk) 11:45, 4 July 2023 (UTC)

I have found solutions, see https://mathlair.allfunandgames.ca/tictactoe.php and https://docplayer.net/storage/70/62426859/1688750420/1P-d-ELuli-47GRD1QIKGA/62426859.pdf, only for L, S, Y pentominos the first player can win (for the U-pentomino, the first player cannot win), and for almost all hexomino (all but the N-hexomino), the first player cannot win, the case for the N-hexomino is still unsolved. 2001:B011:8004:F98F:D41C:CCC9:D3B:31D9 (talk) 16:38, 7 July 2023 (UTC)

Conjecture: There is a value $N$ such that the first player cannot force a win for any $n$-omino with $n\geq N.$  --Lambiam 19:00, 7 July 2023 (UTC)