Could a neural network be capable to diferentiate between two boards of a game? Let's propose, that I can define the state of a board in a board game, with 234 neurons. In theory, could I be able to train a neural network, with 468 inputs (two game boards), and 1 output, to tell me which board state is 'better'? The output should give me ~-1 if the second board is better than the first, ~0 if they are equal, and ~1 if the first board is better than the second.

If yes, what could be the number of ideal neurons on the hidden layers? What could be the ideal number of hidden layers?
For optimal performance, the network complexity should fit the complexity of the game. Since we do not know the latter, your question is not answerable.