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Part of the Monthly Topic Challenge #10: Möbius Strips, Klein Bottles, and other unusual topological surfaces

Warm-up for topological thinking. You are playing tic-tac-toe with the usual 3x3 grid, but instead of putting the grid on flat paper, it is on a torus. The top is glued to the bottom and the left side is glued to the right side. This opens a few more lines that also count as three in a row.

Does the starting player have a winning strategy or is this still a tie if both players play optimally?

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The first player

does have a winning strategy.

On the torus, every pair of points extends to a line of three. And up to symmetry, the first move can be assumed to be the center, and the second move is then either an edge or corner* (as displayed by our flattened table). In either case, after X's second move (there are multiple winning options**, below I've picked one each), things are mostly forced (threats are marked !):

 o..   o..   o!!
 xx!   xxo   xxo
 ...   .!.   .x.
 

 x..   x!.   xx!
 ox.   ox.   ox.
 ..!   ..o   .!o

* Even this can be reduced: rotating the board 45 degrees, the torus-wrapped diagonals become horizontal and vertical lines, and the horizontal and verticals become wrapped diagonals!

** I think in fact

every second move by X can still be pursued to a win.

I've traced through what I think are all those lines, but maybe a slicker proof is available by noting that three points not already in a line complete a line at three distinct places, one for each pair.

Also note that the game cannot end in a tie.

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  • $\begingroup$ Alternately, a strategy stealing argument shows the second player can't force a win, and since the game can't end in a tie, the first player can force a win. $\endgroup$
    – ralphmerridew
    Commented May 6, 2023 at 14:09
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    $\begingroup$ @ralphmerridew I'd be interested in that as a separate answer; it's not immediately obvious to me how a strategy-stealing would work. $\endgroup$
    – Ben Reiniger
    Commented May 6, 2023 at 14:36
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    $\begingroup$ Standard argument by contradiction. Having an extra mark on the board is strictly beneficial. If the game is a win for the second player, then the first player makes a random move on the first move, then follows the second player's strategy. Any time the strategy would ever have first move on that extra square, make another random move. This wins for the first player, contradiction. $\endgroup$
    – ralphmerridew
    Commented May 8, 2023 at 1:19

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