Necklace Games

TL;DR

The paper introduces and analyzes NecklaceNim, a variation of PathNim with end-move options, crucial for solving CircularNim, and provides solutions for infinite families when play involves at least half the stacks.

Contribution

It defines NecklaceNim and solves infinite families of the game for specific conditions, advancing understanding of related combinatorial games.

Findings

Solved infinite families of NecklaceNim for at least half the stacks.

Established the relationship between NecklaceNim and CircularNim.

Provided results critical for solving CircularNim.

Abstract

We define and give results on the game NecklaceNim NN($n$,$k$) which is PathNim PN($n$,$k$) with an additional move allowed on the end vertices. This game arises as a sub-game in the context of solving CircularNim CN($n$,$k$) when $k-2$ consecutive stacks have been depleted, therefore its solution is critical to solving CircularNim. We solve the infinite families of NN($n$,$k$) when play is allowed on at least half the stacks.