Experimental Study of the Game Exact Nim(5, 2)

TL;DR

This paper investigates the properties of the game Exact Nim(5, 2), comparing it with related Nim variants, and provides theoretical and computational insights into its P-positions and Sprague-Grundy values.

Contribution

It offers new theoretical and computational evidence on the relationship between P-positions of nim(6,=2) and nim(6,≤2), advancing understanding of these Nim variants.

Findings

P-positions of nim(5,=2) are closely related to those of nim(6,≤2)

Explicit Sprague-Grundy values known for some cases, open for others

Theoretical and computational evidence supports conjectured relations

Abstract

We compare to different extensions of the ancient game of nim: Moore's nim$(n, \leq k)$ and exact nim$(n, = k)$. Given integers $n$ and $k$ such that $0 < k \leq n$, we consider $n$ piles of stones. Two players alternate turns. By one move it is allowed to choose and reduce any (i) at most $k$ or (ii) exactly $k$ piles of stones in games nim$(n, \leq k)$ and nim$(n, = k)$, respectively. The player who has to move but cannot is the loser. Both games coincide with nim when $k=1$. Game nim$(n, \leq k)$ was introduced by Moore (1910) who characterized its Sprague-Grundy (SG) values 0 (that is, P-positions) and 1. The first open case is SG values 2 for nim$(4, \leq 2)$. Game nim$(n, = k)$, was introduced in 2018. An explicit formula for its SG function was computed for $2k \geq n$. In contrast, case $2k < n$ seems difficult: even the P-positions are not known already for nim$(5,=2)$. Yet, it seems that the P-position of games nim$(n+1,=2)$ and nim$(n+1,\leq 2)$ are closely related. (Note that P-positions of the latter are known.) Here we provide some theoretical and computational evidence of such a relation for $n=5$.