Problem
Draw the top three levels (showing the first two moves) of the game tree for the game of tic-tac-toe (noughts and crosses), and calculate the number of vertices that will appear on the fourth level. You may reduce the size of the tree by taking advantage of symmetries: At the first move, for example, show only three possibilities (the center square, a corner, or a side square) rather than all nine. Further symmetries near the root will reduce the size of the game tree.