1. Introduction

Domineering or Crosscram is a game invented by Goran Andersson and introduced to the public in [1]. Two players, say Vera and Hepzibah, have vertical and horizontal dominoes respectively. They start with a board consisting of some subset of the square lattice and take turns placing dominoes until one of them can no longer move. For instance, the 2 x 2 board is a win for the first player, since whoever places a domino there makes another space for herself while blocking the other player's moves.

A beautiful theory of combinatorial games of this kind, where both players have perfect information, is expounded in [2; 3]. Much of its power comes from dividing a game into smaller subgames, where a player has to choose which subgame to make a move in. Such a combination is called a disjunctive sum. In Domineering this happens by dividing the remaining space into several components, so that each player must choose in which component to place a domino.

Each game is either a win for Vera, regardless of who goes first, or Hepzibah regardless of who goes first, or the first player regardless of who it is, or the second regardless of who it is. These correspond to a value G which is positive, negative, fuzzy, or zero, i.e.,or. (By convention Vera and Hepzibah are the left and right players, and wins for them are positive and negative respectively.) However, we will often abbreviate these values as G = V, H, 1st, or 2nd. We hope this will not confuse the reader too much.

In this paper, we find who wins Domineering on all rectangles, cylinders, and tori of width 2, 3, 5, and 7. We also obtain bounds on boards of width 4, 7, 9, and 11, and partial results on many others. We also comment briefly on toroidal and cylindrical boards.