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On Fraenkel's N-Heap Wythoff's Conjectures
Xinyu Sun1 and Doron Zeilberger2
1Department of Mathematics, Temple University, Philadelphia, PA 19122, USA
xysun@math.temple.edu
2Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA
zeilberg@math.rutgers.edu
Annals of Combinatorics 8 (2) p.225-238 June, 2004
AMS Subject Classification: 91A46, 68R05
Abstract:
The N-heap Wythoff's game is a two-player impartial game with N piles of tokens of sizes . Players take turns removing any number of tokens from a single pile, or removing (a1,..., aN) from all piles --- ai tokens from the i-th pile, providing that and where is the nim addition. The first player that cannot make a move loses. Denote all the P-positions (i.e., losing positions) by and for all . Two conjectures were proposed on the game by Fraenkel [7]. When are fixed, i) there exists an integer N1 such that when ii) there exist integers N2 and such that when and , where , and the golden section.
In this paper, we provide a sufficient condition for the conjectures to hold, and subsequently prove them for the three-heap Wythoff's game with the first piles having up to 10 tokens.
Keywords: Wythoff's game, combinatorial game

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