$\mathcal{P}$ Play in Candy Nim

TL;DR

This paper analyzes Candy Nim, a variant of Nim where players aim to take the last candy and maximize candies taken, providing bounds, strategies, and constructions for optimal play in three-pile positions.

Contribution

It introduces bounds, strategies, and constructions for optimal play in three-pile Candy Nim, advancing understanding of this game variant.

Findings

Bounds on candies obtained in 3-pile $ ext{P}$ positions

Strategies that are provably optimal for certain game families

Construction of games maximizing the loser's candies

Abstract

Candy Nim is a variant of Nim in which both players aim to take the last candy in a game of Nim, with the added simultaneous secondary goal of taking as many candies as possible. We give bounds on the number of candies the first and second players obtain in 3-pile $\mathcal{P}$ positions as well as strategies that are provably optimal for some families of such games. We also show how to construct a game with $N$ candies such that the loser takes the largest possible number of candies and bound the number of candies the winner can take in an arbitrary $\mathcal{P}$ position with $N$ total candies.