We could forbid the players from making "obviously stupid" moves, for different notions of "obviously stupid" (e.g., not making winning moves when they're available, or not blocking winning moves when the other player threatens to win).Two players compete to see who can be the first to get 3 tokens in a line either vertically, horizontally or diagonally.We could stop the game once the winner has been determined (or once it's been determined that the game is a draw).The page I linked to above also lists several other possible answers we could give, under additional constraints: Under game symmetry, the number of possible tic-tac-toe games is only $26\,830$. The difference between the two notions of symmetry described above is board symmetry (which considers the two games above to be different) and game symmetry (which considers them to be the same). This page lists the answer to possible variants of this question. In other words, after every move these two games were in identical positions up to symmetry - it's just that which symmetry this is changes over the course of the game. If you care about the order in which the pieces were played, then a description of one complete tic-tac-toe might look like the board below:Īnd all such descriptions with a three-in-a-row in them are asymmetric: so, because the tic-tac-toe board has eightfold symmetry (four possible rotations which may or may not be followed by a reflection), we can take symmetry into account just by dividing by $8$.īut you might also consider this game to be identical to the game represented byīecause the only distinguishing trait is that on $\bigcirc$'s second move, they played next to $\times$'s first mark, not next to their second mark, even though the two positions created in this way are symmetrical. In one sense, you could just take the answer of $255\,168$ to the previous problem and divide it by $8$, getting $31\,896$.
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