Introduction

Combinatorial game theory [Berlekamp et al. 1982, Conway 1976] has discovered a fascinating array of mathematical structures that provide explicit winning strategies for many positions in a wide variety of games. The theory has been most successful on those games that tend to decompose into sums of smaller games. After assigning a game-theoretic mathematical value to each of the summands, these values are added to determine the value of the entire position. This approach has provided powerful new insights into a wide range of popular games, including Go [Berlekamp and Wolfe 1994], Dots-and-Boxes [Berlekamp 2000a], and even certain endgames in Chess [Elkies 1996].

Some positions on two-dimensional board games breakup into one-dimensional components, and the values of these components often have their own interesting structures. Examples include Blockbusting [Berlekamp 1988], Toads and Frogs [Erickson 1996] and Amazons [Berlekamp 2000b]. The same is true for Konane as suggested by previous analyses of one-dimensional positions [Ernst 1995 and Scott 1999].

Konane is an ancient Hawaiian game similar to checkers. It is played on an 18x18 board with black and white stones placed in an alternating fashion so that no two stones of the same color are in adjacent squares. Two adjacent pieces adjacent to the center of the board are removed to begin the game. A player moves by taking one of his stones and jumping, in the horizontal or vertical direction, over an adjacent opposing stone into an empty square. The jumped stone is removed. A player can make multiple jumps on his turn but cannot change direction mid turn. The first player who cannot make a move loses.

Our goal was to find values for all possible 1 x n Konane positions that consist of three solid patterns each separated by a single space, which we call an “almost solid pattern with two spaces”. We assumed that the positions were far away from the edges of the board so that there was no interference.