A Simple FSM-Based Proof of the Additive Periodicity of the Sprague-Grundy Function of Wythoff's Game

Summary

ABSTRACT. Dress et al. recently proved the additive periodicity of rows of the Sprague—Grundy function of a class of Nim-like games including Wythoff's Game. This implies that the function for each row minus its salt us is periodic and can be computed by a finite state machine.

The proof presented here, which was developed independently and in ignorance of the above result, proceeds in exactly the opposite direction by explicitly constructing a finite state machine with o(n²) bits of state to compute, for all m and fixed n. From this, the periodicity of H and the additive periodicity of G follow trivially.

1. Introduction

Note: The original lecture on this material is available on streaming video at http://www.msri.Org/publications/ln/msri/2000/gametheory/landman/l/.

Wythoff's Game (also called Wythoff's Nim or simply Wyt) is an impartial 2-player game played with 2 piles of counters. Each player may, on their turn, remove any number of counters from either pile, or remove the same number of counters from both piles. The player removing the last counter wins. The set of winning positions (those for which the Sprague-Grundy value equals is well-known. However, no direct formula is known for computing the Sprague-Grundy value G(m,n) of an arbitrary position with m counters in one pile and n in the other; it appears necessary to compute all the G(i, j)-values for smaller games first.

It is clearly impossible to compute the n-th row of (that is, the sequence directly with a finite state machine. One reason is that the values grow without bound, and thus the number of bits needed to represent the value will eventually exceed the capacity of any FSM. It also appears at first that the FSM would have to remember an ever-growing number of values which have already been used. The following lemmas provide bounds on and allow us to overcome these obstacles.