Throughout this paper, the natural numbers = ﹛0, 1, 2, …﹜ are denoted , the set of sequences of natural numbers (integers) are denoted N∞ (Z∞) and the origin is denoted O.

For an introduction to combinatorial game theory, including impartial games, see [WW]. For a detailed description of misère quotient monoids, see [PS]. An algorithm for computing quotient monoids is found in [W].

1. Introduction

Traditionally, normal play impartial games and misère play impartial games are thought to differ by their respective winning conditions, that is, the goal of normal play is to make the final move to the empty game, whereas the goal of misère play is to force the opponent to make the final move. Both versions end when the last bean is removed. Here we present a different perspective, that normal play and misère play differ in the set of positions which are legal, so that both end with the same winning condition: having your opponent unable to move to a legal position.

Also, the traditional description of impartial games relies on the notion of the sum of two games and the theory is typically presented in terms of arbitrary sums of games. Here we adopt a notation that includes arbitrary sums of games under a particular ruleset as a set of lattice points ⊆ N∞. Hence we need not talk about sums of positions; all discussion will be localized to legal moves from a particular lattice point, since the notion of sum is inherent in the lattice point definition of a position.

Our basic framework is to play on several (but a finite number of) heaps of beans, where the rules allow a heap to be replaced by specified finite multisets of heaps of smaller sizes.

2. Heap games

Our definitions are motivated by the game of Nim, in which the set of all legal positions is generated by the set of heaps of various sizes.