Solving narrow Konane boards

Abstract

In this paper we investigate the game of Konane, using Combinatorial Game Theory and game-specific solving strategies. We focus on narrow rectangular boards ( $\mathit{m}\times \mathit{n}$ boards with $\mathit{m}⩽4$ ). These are dubbed as Linear Konane, Double Konane, Triple Konane, and Quadruple Konane, respectively.

The initial board contains black and white stones in a checkered pattern, with a gap of two adjacent empty squares to enable moving. Only capture moves are possible. Depending on the exact location of the initial gap (the setup) we have four classes of initial Konane boards (two for Linear Konane), namely all combinations of a horizontal or vertical setup in the middle of the board or at a corner.

For solving narrow Konane boards two notions proved very useful. First, we define moves that cannot be prevented by the opponent as safe moves of a player. Second, two fragments are independent if there is no way they can ever interact.

Using these two notions Linear and Double Konane have been completely solved. Triple Konane was solved except for the horizontal corner setup. For Quadruple Konane only the vertical setup in the middle of the board was solved.