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Published online by Cambridge University Press: 29 May 2025
ABSTRACT. Amazons is a young “real world” game that fulfills all defining constraints of combinatorial game theory with its original rule set. We present a program to evaluate small Amazons positions with a given number of amazons on game boards of restricted sizes with canonical combinatorial game theory values. The program does not use an analytical approach, instead it relies on exhaustive search in a bottom-up strategy. Here it is applied on all positions on game boards which fit into an underlying game board of size 11 by 2.
The results show that even under these restrictions Amazons offers a wide spread of game theoretic values, including some very interesting ones. Also the canonical forms of the values can be very complicated.
1. Introduction
“The Game of the Amazons” — or simply “Amazons” — is a relatively new star on the sky of abstract strategic two-person games, invented in 1988 by Argentinian Walter Zamkauskas. Compared with other non-classical games it seems to be well done — easy rules, many interesting choices for each move, a big range of different tactics and strategies to employ, challenging even after many games played. It has yet to prove if it can stand the comparison to classics like chess and go.
Amazons has already built a solid base of players and followers. It has especially produced interest in programmers, leading to several Amazons programming competitions including a yearly world-championship and a tournament at the Mind Sports Olympiad 2000, see for example [Iida and Müller 2000] and [Hensgens and Uiterwijk 2000]. Already there are about a dozen different Amazons programs competing at these tournaments. Some of these programs play strong enough to beat average human players easily.
In contrast to most games of this type combinatorial game theory can be applied directly to Amazons without any changes to the rules. Amazons also often decomposes into at least two independent subgames when the endgame is played out, making the application of combinatorial game theory worthwhile.
The first analysis of Amazons with the means of combinatorial game theory was done by E. Berlekamp, who looked at positions with one amazon per player on boards of size 2 by n [Berlekamp 2000]. Berlekamp calculated only the thermographs for these positions, but even these proved to be quite difficult to analyze.
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