mmarkley@microsoft.com writes: I seem to remember from way back in high school that the number of potential moves by the third set of moves is on the order of billions of legal moves. The number of moves in a given chess position is less than 64 (number of starting squares) times 64 (number of destination squares) x 4 [number of ways a pawn can promote]. Thus we get the bound 16, 384 [which can be easily improved] which is way less than "billions of possible moves". The same computation shows that the number of possible games of length n grows at worst expoentially pace mr markley. The right way to think about this is to get sharp upper bounds rather than attempt a precise calculation. A crude upper bound would be longerst possible game is about 6000 moves [using the 50 move rule]. At most 2**16 mves per position so at most 10**[192 * 10**6] games. I'm sure that sharper estimates are readily available.