[Math] What are the symmetries of a tic tac toe game board

Question

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!

Answer 1

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.