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) :
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/.
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.