# What are the symmetries of a tic tac toe game board?

What are the symmetries of the tic tac toe board game? Ie, what are the ways you can rotate, reflect, and/or flip the tic tac toe board, such that the next best move to a board(after it was rotated, reflected, etc) is still the next best move after the board was rotated/reflected/fipped? How would i also construct a group multiplication table for these symmetries?

Thank You!

-
Please show some work. –  Henning Makholm Oct 17 '11 at 0:17

This may be a more subtle question than it seems at first sight.

The easy answer might be that the board is a $3\times 3$ square and so you are looking at the symmetry group of a square.

However, the number of possible different games is known to be 255,168 ignoring symmetry and 26,830 taking symmetry into account. Surprisingly, the latter number is less than one-eighth of the former. The way I once tried to explain this was

• the first diagram below is equivalent to the second using a reflection in the line between the top right and bottom left, so they can both be considered as being the third;
• therefore the fourth must be equivalent to the fifth, since they are both essentially the sixth, which is simply the third with two extra moves.

-
On the other hand, if one simply numbers the moves consecutively, the first and second are equivalent under a reflection, but the fourth and fifth aren’t: in the fourth move $4$ is adjacent to move $1$, while in the fifth it’s adjacent to move $3$. Under this stricter (and in my view more reasonable $-$ the order of the moves is part of the game) notion of equivalence I believe that you do get the expected number of different games. –  Brian M. Scott Oct 17 '11 at 0:57
@Brian. Indeed. You get 31,896 possible games (one eighth of 255,168) with your rather less exciting definition of symmetry. –  Henry Oct 17 '11 at 0:59
May you live in interesting ti- definitions! :-) –  Brian M. Scott Oct 17 '11 at 1:01