What are the symmetries of a tic tac toe game board? What are the symmetries of the tic tac toe board game? Ie, what are the ways you can rotate,
reflect, and/or flip the tic tac toe board, such that the next best move to a board(after it was rotated, reflected, etc) is still the next best move after the board was rotated/reflected/fipped? How would i also construct a group multiplication table for these symmetries? 
Thank You!
 A: This may be a more subtle question than it seems at first sight.  
The easy answer might be that the board is a $3\times 3$ square and so you are looking at the symmetry group of a square.  
However, the number of possible different games is known to be 255,168 ignoring symmetry and 26,830 taking symmetry into account.  Surprisingly, the latter number is less than one-eighth of the former.   The way I once tried to explain this was

  
*
  
*the first diagram below is equivalent to the second using a reflection in the line between the top right and bottom left, so they
  can both be considered as being the third;
  
*therefore the fourth must be equivalent to the fifth, since they are both essentially the sixth, which is simply the third with two
  extra moves.
  


A: Why isn't the total number of possible games 362880? That would be 9*8*7*6*5*4*3*2.
A: I once made a ticktacktoe AI, and in order to keep my potential outcomes small I rotated the board. When the user would make a choice I rotated the placements on the board so that the mouse would be over one of two options that the AI was concerned with. If AI takes mid he now thinks the only choices for the user are bottom-right and bottom-middle. If you chose top I would pass the second choice bot-mid and do a 180 flip to the board so you would choose what appears to be top-mid. I hope this helps.
