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Consider a $3 \times 3$ Tic-Tac-Toe board with opposite sides identified in opposite orientation. We play Tic-Tac-Toe in the Real Projective Plane.

More precisely, consider a $3 \times 3$ Tic-Tac-Toe board on the unit square $[0,1]^2$. We glue the boundary of the square in the following way to create the Real Projective Plane : $(x,0) \sim (1-x,1)$ and $(0,y) \sim (1,1-y)$. That way, we have created a $3 \times 3$ Tic-Tac-Toe board on the Real Projective Plane. We still have $9$ cells, but we have more possible ways to form winning patterns since we can go beyond the sides of the planar board.

A way to visualize the game is to create copies of the board (with appropriate rotations) around a planar board. Here is what it would look like for the Klein Bottle (picture taken from Mathematics Illuminated) :

Tic-Tac-Toe on a Klein Bottle

It seems that any three moves is a first-player win in the Real Projective Plane game. Is there a smart way to see it (except check all the $84$ cases) ? Is this result related in any way to the topology (or properties) of the Real Projective Plane ?

There is an action by horizontal and vertical translations (with appropriate reflection when a line goes through a side) on the board from the group $\Bbb Z_6 \times \Bbb Z_6$ that preserves winning patterns. We could also consider $\pi/2$ rotations through the center of the board. Looking at the order of the symmetry group, there must be some inequivalent three-element subsets of the board. Maybe there is a smart way to identify the orbits ?

This is related to Prove that a game of Tic-Tac-Toe played on the torus can never end in a draw. (Graph theoretic solutions only.). Similar arguments can probably be used to show that our game cannot end in a draw, if that's helpful in any way.

This problem comes from https://concretenonsense.wordpress.com/2008/04/15/topological-tic-tac-toe-1-the-torus/ and https://concretenonsense.wordpress.com/2008/04/17/topological-tic-tac-toe-2-other-surfaces/.

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  • $\begingroup$ I am not quite sure what you mean here - are you working with eighteen cells or identifying each cell with its reverse? Also it is possible to identify three moves by the first player which don't make a line, but where the second player is able to make a line. So are you assuming normal sequential play where the third move by the first player has to make a line (first priority) or block a threat (second priority) before considering other options? $\endgroup$
    – Mark Bennet
    Mar 9, 2019 at 10:49
  • $\begingroup$ @MarkBennet : I think it means identify as in topology, so you'll think of the rightmost column as being adjacent to the leftmost one, but reversed (same for top and bottom rows) $\endgroup$
    – Maxime Ramzi
    Mar 9, 2019 at 10:59
  • $\begingroup$ @Max Then you get lines which are six cells long - does a win require three adjacent cells in a line or just three in a row? And doing it that way also seems to mean that two cells can lie on more than one common line. $\endgroup$
    – Mark Bennet
    Mar 9, 2019 at 11:03

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The claim made in the Concrete Nonsense blog is not actually true. The neighbourhoods of corner cells on the $\mathbb{RP}^2$ tic-tac-toe board are different from that of the centre – corner cells are diagonally adjacent to themselves – and this prevents using the translation arguments from the toroidal board. Indeed, I found $7$ cases, up to rotation and reflection, where the first three first-player moves don't lead to a win: However, the same peculiar properties of $\mathbb{RP}^2$ allow the first player to win in two moves. The first move is at the centre and the second move is at any unoccupied corner.

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    $\begingroup$ I guess the visualization by adding boards is misleading. A winning pattern is one that can be made into a usual winning pattern (horizonal/vertical/diagonal line) by "translating" the board along glued sides. The X's and O's move along with the board. If you have only two X's on your board, there is no way you can translate it to make it three, even though the visualization would make it think so. But it seems that the blog post was indeed wrong. $\endgroup$
    – Desura
    Mar 9, 2019 at 15:18
  • $\begingroup$ @Desura You cannot translate the real projective plane because both directions are anti-parallel. You can translate a Klein bottle, a torus, because the copies line up in parallel strips. Shifting the RPP would introduce gaps. $\endgroup$
    – Parcly Taxel
    Mar 9, 2019 at 18:17

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