An Amalgamation Nim with Restriction

TL;DR

This paper introduces a variant of Nim called Amalgamation Nim, where players can combine two piles under specific conditions, and provides a formula to identify P-positions for this game variant.

Contribution

The paper presents a new rule for amalgamating piles in Nim and derives a formula for P-positions under this rule, which was previously unknown.

Findings

A condition for amalgamating two piles based on their size.

A formula describing the set of P-positions in the new game.

Enhanced understanding of strategic positions in Amalgamation Nim.

Abstract

In an amalgamation Nim, players are allowed to use a move from the traditional form of Nim or to amalgamate two piles when they are not empty. No formula that describes the set of P-positions of Amalgamation Nim is known. The author gives a condition on the amalgamation of two piles. Players can amalgamate two piles only when the number of stones in these two piles is equal to or more than 2. Then, we get a formula that describes the set of P-positions for this variant of amalgamation Nim.