# Tic-Tac-Toe on the Real Projective Plane is a trivial first-player win in three moves

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.

• 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? Mar 9 '19 at 10:49
• @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) Mar 9 '19 at 10:59
• @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. Mar 9 '19 at 11:03

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.