More on the Sprague–Grundy function for Wythoff's game

Abstract

ABSTRACT. We present two new results on Wythoff's Grundy function Ᏻ. The first one is a proof that for every integer g 0, the g-values of Ᏻ are within a bounded distance to their corresponding 0-values. Since the 0-values are located roughly along two diagonals, of slopes and 1 , the g-values are contained within two strips of bounded width around those diagonals. This is a generalization of a previous result by Blass and Fraenkel regarding the 1-values.Our second result is a convergence conjecture and an accompanying recursive algorithm. We show that for every g for which a certain conjecture is true, there exists a recursive algorithm for finding the n-th g-value in O.log n/ arithmetic operations. Our algorithm and conjecture are modifications of a similar result by Blass and Fraenkel for the 1-values. We also present experimental evidence for our conjecture for small g.

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

“…of[5]. Based on computer experiments, we conjecture that this asymptotic behavior holds in F-Wythoff.Conjecture 18.…”

“…In standard Wythoff, for each j ∈ N 0 , we know that lim n→∞ The following theorem of Nivasch [5] reduces the above conjecture to the statement that the positive difference between d j n and n is bounded by some constant c j independent of n.…”

“…Theorem 19 (Nivasch [5]). Let A := {a n : n 0} ⊂ N 0 be a strictly increasing sequence and B := {b n : n 0} ⊂ N 0 be a sequence of distinct elements.…”

“…11 11 10 9 12 8 15 13 0 2 3 16 14 7 10 10 11 8 9 7 14 15 3 13 5 6 16 17 9 9 8 11 10 12 13 1 2 6 7 5 3 4 8 8 9 10 7 11 0 12 4 5 6 13 2 1 7 7 6 5 8 9 1 10 11 4 2 3 0 15 6 6 7 4 5 0 2 3 10 12 1 15 13 8 5 5 4 7 6 3 8 2 1 0 13 14 15 9 4 4 5 6 1 2 3 0 9 11 12 7 8 10 3 3 2 0 4 1 6 5 8 7 10 9 12 11 2 2 3 1 0 6 7 4 5 10 11 8 9 14 1 1 0 3 2 5 4 7 6 9 8 11 10 13 0 0 1 2 3 4 5 6 7 8 9 10 11 12 j/i 0 1 2 3 4 5 6 7 8 9 10 11 12 In Wythoff's game, it is known that the set of P-positions is {( φn , φ 2 n ) : n 0}, where φ is the golden ratio, up to reordering of the coordinates [6]. Moreover, the Sprague-Grundy values of standard Wythoff have been studied extensively in [1], [4], and [5]. It is known that every column of Sprague-Grundy values is ultimately additively periodic [4].…”

“…It is known that every column of Sprague-Grundy values is ultimately additively periodic [4]. Additionally, every row, column, and diagonal of Sprague-Grundy values contains each nonnegative integer exactly once [5,1]. In [5], it is shown that for a fixed g, the positive difference between the nth g-position and the nth 0-position is bounded by a constant depending solely on g, where the sequences are in order of increasing first coordinates with the smaller pile size in the first coordinate.…”