By Xinyu Sun and Doron Zeilberger

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Appeared in Annals of Combinatorics 8(2004), 225-238.

Written: Nov. 13, 2003.

The most famous Combinatorial (impartial) game is Nim. It is also trivial, especially the 2-heap case. In 1907 Wythoff suggested a very interesting variation on 2-heap Nim, that added an extra move: take the same amount from both piles. A few years ago, the great Combinatorial Gamer, Aviezri Fraenkel, suggested a natural generalization of Wythoff's game to many piles, where the moves are given by "Losing Nim-Positions", and made interesting conjectures about the losing positions. Here we prove Fraenkel's conjectures for three piles, with the smallest pile of size less than eleven.

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