title: TicTacToe strategic reduction
body: <p>I decided to write a small program that solves TicTacToe in order to try out the effect of some pruning techniques on a trivial game.  The full game tree using minimax to solve it only ends up with 549,946 possible games.  With alpha-beta pruning, the number of states required to evaluate was reduced to 18,297.  Then I applied a transposition table that brings the number down to 2,592.  Now I want to see how low that number can go.</p>
  
  <p>The next enhancement I want to apply is a strategic reduction.  The basic idea is to combine states that have equivalent strategic value.  For instance, on the first move, if X plays first, there is nothing strategically different (assuming your opponent plays optimally) about choosing one corner instead of another.  In the same situation, the same is true of the center of the walls of the board, and the center is also significant.  By reducing to significant states only, you end up with only 3 states for evaluation on the first move instead of 9.  This technique should be very useful since it prunes states near the top of the game tree.  This idea came from the GameShrink method created by a group at CMU, only I am trying to avoid writing the general form, and just doing what is needed to apply the technique to TicTacToe.</p>
  
  <p>In order to achieve this, I modified my hash function (for the transposition table) to enumerate all strategically equivalent positions (using rotation and flipping functions), and to only return the lowest of the values for each board.  Unfortunately now my program thinks X can force a win in 5 moves from an empty board when going first.  After a long debugging session, it became apparent to me the program was always returning the move for the lowest strategically significant move (I store the last move in the transposition table as part of my state).  Is there a better way I can go about adding this feature, or a simple method for determining the correct move applicable to the current situation with what I have already done? </p>