1. Introduction

A natural enumeration question for combinatorial games is: “How many legal positions are possible in a game?” Surprisingly, few have actually considered this problem. Farr [7; 8], and Tromp and Farnebäck [20] consider the problem of “counting the number of end positions in GO.” Similar enumeration questions are addressed by Hetyei [12], who analyses a game where the number of P-positions (second player win positions) of length n is related to the n-th Bernoulli number of the second kind, and in [17], where it is shown that for the game of TIMBER, on paths, the number of P-positions of length n is related to the Catalan and Fine numbers.

In Section 3, we enumerate the positions of several well-known games. A natural subset of combinatorial games, which we call placement games, are those that consist of placing pieces on a board until the board “fills” and there are no further moves. For each game, except NOGO, we find an auxiliary graph for which a position in the game corresponds to an independent set in the auxiliary graph.

In Section 4, we exhibit bijections between the games with identical generating functions.

2. Background

A placement game can be abstractly represented as a game on a graph, with the following properties.

• • The game begins on a graph that contains no pieces.

• • A move is to place a piece on one (or more) vertices subject to the rules of the particular game.

• • The rules must imply that if a piece can be placed in a certain position on the board then it was legal to place it in that position at any time earlier in the game.

• • Once played, a piece remains on the graph; it is never moved or removed from the graph.

Placement games were first identified during the seminar which led to this paper and have become of interest because of their properties; see [13; 6; 5; 16].