Experiments in Computer Amazons

Summary

ABSTRACT. Amazons is a relatively new game with some similarities to the ancient games of chess and Go. The game has become popular recently with combinatorial games researchers as well as in the computer games community. Amazons combines global full-board with local combinatorial game features. In the opening and early middle game, the playing pieces roam freely across the whole board, but later in the game they become confined to one of several small independent areas.

A line segment graph is an abstract representation of a local Amazons position. Many equivalent board positions can be mapped to the same graph. We use line segment graphs to efficiently store a table of defective territories, which are important for evaluating endgame positions precisely. We describe the state of the art in the young field of computer Amazons, using our own competitive program Arrow as an example. We also discuss some unusual types of endgame and zugzwang positions that were discovered in the course of writing and testing the program.

1. Introduction

The game of Amazons was invented by Walter Zamkauskas. Two players with four playing pieces each compete on a 10 x 10 board. Figure 1 shows the initial position of the game. The pieces, called queens or amazons, move like chess queens. After each move an amazon shoots an arrow, which travels in the same way as a chess queen moves. The point where an arrow lands is burned off the playing board, reducing the effective playing area. Neither queens nor arrows can travel across a burned off square or another queen. The first player who cannot move with any queen loses.

Amazons endgames share many characteristics with Go endgames, but avoid the extra complexity of Go such as ko fights or the problem of determining the safety of stones and territories. Just like Go, Amazons endgames are being studied by combinatorial games researchers. Berlekamp and Snatzke have investigated play on sums of long narrow n x 2 strips containing one amazon of each player [1; 15]. Even though n x 2 areas have a simple structure, sum game play is surprisingly subtle, and full combinatorial game values become very complex.