The Game of Amazons

Amazons is a board game in which each of the two players controls a few pieces called "Amazons".  Traditional games begin with four Amazons of each color in a defined starting position on a 10 x 10 board; shorter games are sometimes played with each player having three Amazons on an 8 x 8 board. Each Amazon moves like a chess queen: any desired distance in a straight line in any of 8 directions. After landing on an empty square, the Amazon must complete her move by shooting a flaming arrow, which also moves in a straight line in any of the 8 directions. When the arrow lands, it burns that square off of the board. This is conventionally denoted placing an immobile Go stone on each burned-out square. Neither Amazons nor arrows can move onto or over burned-out squares. Eventually so many squares are burned out that some player, at his turn, is unable to move. The game then ends and that player loses.

In most games, the burned-out squares eventually partition the board into several disjoint regions. We show how combinatorial game theory allows us to evaluate each region, and to sum these values to determine optimal strategies for playing specific endgames.