A survey about Solitaire Clobber

Summary

Solitaire Clobber is a one-player variant of the 2-player board game Clobber introduced by Albert et al. in 2002. According to simple rules, the objective of Solitaire Clobber is to capture the maximum number of stones from a given graph. Two versions of Solitaire Clobber were recently investigated: a partisan and an impartial one. In this survey, we give an overview of the major results about Solitaire Clobber, more especially about the impartial version. In particular, the game is considered on grids, trees, and hypercubes. Two new results are provided: when playing on a tree, we show that the minimum number of remaining stones can be computed in polynomial time. We also assert that any game position on a “large” grid can be reduced to 1 or 2 stones. Note that in each part of this survey, we propose several open problems related to Solitaire Clobber.

In 2001, Albert, Grossman, Nowakowski and Wolfe investigated a new 2-player partisan game called Clobber; they developed the first results in [Albert et al. 2005]. In terms of game values, it turns out that Clobber is difficult even when played on basic positions. This complexity explains the author’s motivation for studying Clobber. The description of the game follows below.

Black and white stones are placed on the vertices of an undirected graph, at most one per vertex. The first player moves only black stones and the second player the white ones. A player moves by picking up one of his stones and “clobbering” an adjacent stone of the opposite color (vertically or horizontally). The clobbered stone is deleted and replaced by the one that was moved. The last player to move wins.

Clobber is usually played on a grid where the initial position is the one of a checkerboard, as depicted by Figure 1.

In the last few years, several events were organized around Clobber like the first international Clobber tournament. It was held at the 2002 Dagstuhl seminar on algorithmic and combinatorial game theory (see the report in [Grossman 2004]). Since 2005 the game has been one of the events of the Computer Olympiad.