On the Sprague-Grundy Values of the $\mathcal{F}$-Wythoff Game

Abstract

We examine the Sprague-Grundy values of $\mathcal{F}$-Wythoff, a restriction of Wythoff's game introduced by Ho, where the integer ratio of the pile sizes must be preserved if the same number of tokens is removed from both piles. We answer two conjectures raised by Ho. First, we show that each column of Sprague-Grundy values is ultimately additively periodic. Second, we prove that every diagonal of Sprague-Grundy values contains all the nonnegative integers. We also investigate the asymptotic behavior of the sequence of positions attaining a given Sprague-Grundy value.

The Sprague–Grundy function for some nearly disjunctive sums of Nim and Silver Dollar games

“…Table 1. (1,4), (2,3), (6,9), (7,10), (8,11), (4k + i, (1,5), (2,4), (3,7), (8,10), (9,11), (4k + i, 4k…”

“…But (10, 10) appears in T 0 so step (2b) fails. By increasing z to 1 we have (10,11) appearing in T 2 , so step (2b) fails. Then we increase z to 2, giving (10,12) and satisfying step (2).…”

“…Ultimately additive periodicity has been found for the Wythoff game and some of its variants [7,10,11,14]. In these variants, the players alternately move from a position (a, b), following some given rules.…”

Sprague-Grundy Values of the $\mathcal{R}$-Wythoff Game

“…As a tool for proving Theorem 7, we will use the following characterization of the sequence T g due to Jiao [4].…”

“…Proposition 4 (Lemma 8 in [4]). Every T g consists exactly of the positions having Sprague-Grundy value g if and only if every T g satisfies 1.…”