On Fraenkel’s N-Heap Wythoff’s Conjectures

The N-heap Wythoff’s game is a two-player impartial game with N piles of tokens of sizes A1, . . . , AN , A1 ≤ ·· · ≤ AN . 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 0 ≤ ai ≤ Ai, ∑i=1 ai > 0 and a1 ⊕ ·· · ⊕ aN = 0, where ⊕ is the nim addition. The first player that cannot make a move loses. Denote all the P-positions (i.e., losing positions) by (A1, . . . , AN−2, AN−1 n , A N n ), A N−2 ≤ AN−1 n ≤ An and AN−1 n < AN−1 n+1 for all n ≥ 1. Two conjectures were proposed on the game by Fraenkel [7]. When A1, . . . , AN−2 are fixed, i) there exists an integer N1 such that when n > N1, An = A N−1 n + n. ii) there exist integers N2 and α2 such that when n > N2, AN−1 n = bnφc+ εn + α2 and An = AN−1 n + n, where −1 ≤ εn ≤ 1 and φ = (1+ √ 5)/2, 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.