1. Introduction and Background

We assume the reader is familiar with the first volume of Winning Ways [Berlekamp et al. 1982], including Conway's axiomatization of G, the group of partesan games under addition, which can also be found in [Conway 1976]. I now call the elements of this group traditional games. Each of the traditional games considered in this paper has only a finite number of positions. The identity of G is the game called 0, which is an immediate win for the second player. We investigate some interesting extensions of the group G to a bigger semi-group, S, generated by some new elements which are idempotents in the sense that each of them satisfies the equation G + G = G. We also present an addition table for these idempotents.

Some of these idempotents have long been well-known in other contexts. The newer ones all fall into a class I have been calling enriched environments. A companion paper [Berlekamp 2002] shows how these idempotents prove useful in solving a particular hard problem involving a gallimaufry of checkers, chess, domineering and Go.

We begin with a review of definitions, modified slightly to fit our present purposes.

Moves. In Go, a move is the change on the board resulting from the act of a single player. In chess, this is commonly called a ply, and the term move is used to describe a consecutive pair of plys, one by White and one by Black.