I am trying to calculate how many unique states are possible to be in during a game of hex.
The upper bound for an $n\times n$ board is $3^{n^2}$. This is ignoring gameplay and simply considering that each space could be any of $3$ states. That number contains many states (all black, all white, etc..) that are impossible to reach in any actual game.
In a real game, the constraint will be added that the number of black spaces can be at most one greater than the number of white spaces. I cannot think of a way to quantify that number of states. The number is also reduced by winning states for either problem, which effectively stop any subsequent paths.
I have considered that the first move can be any of $n^2$ spaces, the second $n^2-1$... This gives $(n^2)!$ number of states which is even bigger than my upper bound because of duplicate configurations happening on different paths. I don't care about the path to the state, just what the board looks like.
How many board configurations are really possible on an $n\times n$ board?