Mis\`ere Games and Mis\`ere Quotients

TL;DR

This paper provides an introduction to misère quotients in impartial combinatorial games, covering foundational concepts, key theorems, and recent research directions in the field.

Contribution

It offers a comprehensive overview of misère quotient construction, including proofs of important theorems and discussion of current open problems.

Findings

Proof of the Guy--Smith--Plambeck Periodicity Theorem

Introduction to basic misère quotient construction

Discussion of recent results and open problems

Abstract

These lecture notes are based on a short course on mis\`ere quotients offered at the Weizmann Institute of Science in Rehovot, Israel, in November 2006. They include an introduction to impartial games, starting from the beginning; the basic mis\`ere quotient construction; a proof of the Guy--Smith--Plambeck Periodicity Theorem; and statements of some recent results and open problems in the subject.