Symmetry and Tic Tac Toe

Question

How would one find the number of unique tic tac toe states taking into account rotational equivalence between states?

Answer 1

This question can be understood in many different ways. There are a lot of ways to define a "tic tac toe-game". Below is a little summary of possible assumptions you can make. $$\text{What is a unique move?}$$ (1) Choice of first player is distinct and every board position is distinct. (2) Choice of first player is arbitrary and every board position is distinct. (3) Choice of first player is arbitrary and board orientation is arbitrary, but once the board is oriented every position is distinct (4) Choice of first player is arbitrary and only distinct player choices are distinct moves. $$\text{What moves are allowed?}$$ (1) Any open square. (2) Players must make three in a row if possible. (3) Players must make three in a row if possible, otherwise must make a blocking move if possible. $$\text{When is a game over?}$$ (1) When either player makes three in a row or the board is full. (2) When the outcome of the game is fully determined. $$\text{The most logical answer would be 23,129 possibilities.}$$ For this is the number of games if two games are the same when the players faced the same choices at each move of the game; i.e. considering symmetry at each step of the game. Also, the players don't take meaningless steps and the game is ended when 3 in a row is achieved by one of them. Look at http://www.mathrec.org/old/2002jan/solutions.html if you want to see the full explanation of the computations and the game outcomes overall.