Three-pile Sharing Nim and the quadratic time winning strategy

TL;DR

This paper introduces a variant of 3-pile Nim with topping-up moves, revealing palindromic Sprague-Grundy values, and provides a quadratic-time formula for identifying winning positions and analyzing nim-sequences.

Contribution

It presents a new Nim variant with topping-up moves, discovers palindromic Sprague-Grundy columns, and derives a quadratic-time method for computing P-positions.

Findings

Palindromic structure in Sprague-Grundy values

Quadratic-time formula for P-positions

Analysis of nim-sequence periodicity

Abstract

We study a variant of 3-pile Nim in which a move consists of taking tokens from one pile and, instead of removing then, topping up on a smaller pile provided that the destination pile does not have more tokens then the source pile after the move. We discover a situation in which each column of two-dimensional array of Sprague-Grundy values is a palindrome. We establish a formula for P-positions by which winning moves can be computed in quadratic time. We prove a formula for positions whose Sprague-Grundy values are 1 and estimate the distribution of those positions whose nim-values are g. We discuss the periodicity of nim-sequences that seem to be bounded.