Narrow misère Dots-and-Boxes

Summary

We study misère Dots-and-Boxes, where the goal is to minimize score, for narrow boards. In particular, we characterize the winner for 1xn boards with an explicit winning strategy for the first player with a score of [(n -1)/3]. We also give preliminary results for 2xn and for Swedish 1xn (where the boundary is initially drawn).

Recall the classic children’s game Dots-and-Boxes [Berlekamp et al. 2003]. We start with an m x n square grid of dots. Players alternate drawing individual edges of the grid. If a player completes a box of the grid, s/he gets a point and must draw another edge; this process can repeat several times within a single turn. The game ends when all edges have been drawn, i.e., when all mn boxes have been completed. In normal Dots-and-Boxes, the player to receive the most points wins. In misère Dots-and-Boxes, the player to receive the fewest points wins. A draw (tie) occurs when mn is even and the players complete the same number of boxes.

Normal Dots-and-Boxes endgames are known to be NP-hard; see [Demaine and Hearn 2009]. In addition, no winning strategies are known when m or n is sufficiently large. To our knowledge, even the 1xn case is open for arbitrary n. On the other hand, misère Dots-and-Boxes may be easier to analyze.