I have recently been working out the full game tree for tic-tac-toe, just for fun. I am using the well known equivalence relation of rotations/reflections to simplify this tree in the standard way (which begins by noting that there are only 3 opening moves: edge, corner, and center). I actually could not find an image of the full game tree for tic-tac-toe, so if anyone can provide a link I'd appreciate that.
This work motivates the following;
Let there be given an N x N grid; and let $m$ be a natural number.
Let $I$ be the set of all possible ways to place $m$ copies of the letter $X$ in the grid, and $m$ copies of the letter $O$ in the grid (with the restriction that we can only place one letter per space of the grid; in other words, just imagine playing a game of tic-tac-toe on an N x N board for an even number of moves).
How many ways are there to completely break the rotational/reflection symmetry of an N x N grid by placing $m$ copies of $X$ and $m$ copies of $O$?