Nim is Easy, Chess is Hard - But Why??

In nim, a finite number of piles of finitely many tokens is given. Two players alternate in selecting a pile and removing from it any positive number of tokens, possibly the entire pile. The player first unable to move loses and the opponent wins. Nim is easy. Why? Because it has an easy strategy: write the number of tokens in each pile in binary, and “add” them without carry, an addition also known as XOR (Exclusive Or). If the sum is 0, you better be a gentle(wo)man and offer your opponent to move first, because you can win as second player. If the sum is nonzero you can move first and make it 0, thus winning. Chess is difficult. Why? Lewis Dartnell wrote in “Practice makes perfect”, Plus, Issue 28 January 2004, comparing chess with nim: “Chess, however, is almost inconceivably more complex, and the pieces can be arranged on the 64 squares of the board in 1044 distinct ways. One mathematician has calculated that there are about 10 50) different legal games, which is far more than the number of particles in the entire visible universe. This is effectively an infinite number of permutations, and so in all practical senses it is impossible to play chess perfectly.” A similar reason is given by Marianne Freiberger, in her review of “Luck, logic and white lies”, Plus, Issue 35 May 2005: “In combinatorial games such as chess, the number of possible combinations of moves is astronomical, meaning that a complete analysis is totally unfeasible.” In J. Recreational Math., 19:2, pp. 119-125, 1987, Steven Goldberg wrote, “There is a way to play chess so that you never lose: the good news is that this has been proved. The bad news is that all the computers in the world will never be able to discover it”. The latter claim is based on the statement that “the number of possible moves in chess has been estimated to be approximately 12× 10”. Incidentally, the difference in the estimates of Dartnell and Goldberg is in itself super-astronomical. But the former refers to the number of games, whereas we are interested in the number of moves. Therefore we stick with Goldberg’s estimate. ∗[email protected] http://www.wisdom.weizmann.ac.il/~fraenkel