# Partition games

## TL;DR

This paper introduces the class of 2-player partition games called CUT, analyzes their Sprague-Grundy sequences, and develops computational methods to determine their periodicity and structure.

## Contribution

It systematically studies the subclass 'break-without-take' and proves periodicity properties for several rulesets, extending previous work on take-and-break games.

## Key findings

- Several rulesets have periodic or arithmetic periodic Sprague-Grundy sequences.
- Developed a computational testing condition for analyzing additional games.
- Proved periodicity directly for infinite classes of partition games.

## Abstract

We introduce CUT, the class of 2-player partition games. These are NIM type games, played on a finite number of heaps of beans. The rules are given by a set of positive integers, which specifies the number of allowed splits a player can perform on a single heap. In normal play, the player with the last move wins, and the famous Sprague-Grundy theory provides a solution. We prove that several rulesets have a periodic or an arithmetic periodic Sprague-Grundy sequence (i.e. they can be partitioned into a finite number of arithmetic progressions of the same common difference). This is achieved directly for some infinite classes of games, and moreover we develop a computational testing condition, demonstrated to solve a variety of additional games. Similar results have previously appeared for various classes of games of take-and-break, for example octal and hexadecimal; see e.g. Winning Ways by Berlekamp, Conway and Guy (1982). In this context, our contribution consists of a systematic study of the subclass `break-without-take'.