Solitaire Clobber

TL;DR

This paper introduces Solitaire Clobber, a one-player variant of the two-player game Clobber, analyzing its complexity and reduction strategies for various board configurations.

Contribution

It provides new complexity results and reduction bounds for Solitaire Clobber on different board types and configurations.

Findings

Checkerboard single-row/column can be reduced to about n/4 stones.

Two-row/two-column checkerboard can be reduced to one or two stones depending on the number of stones.

Deciding if an arbitrary configuration can be reduced to a single stone is NP-complete.

Abstract

Clobber is a new two-player board game. In this paper, we introduce the one-player variant Solitaire Clobber where the goal is to remove as many stones as possible from the board by alternating white and black moves. We show that a checkerboard configuration on a single row (or single column) can be reduced to about n/4 stones. For boards with at least two rows and two columns, we show that a checkerboard configuration can be reduced to a single stone if and only if the number of stones is not a multiple of three, and otherwise it can be reduced to two stones. We also show that in general it is NP-complete to decide whether an arbitrary Clobber configuration can be reduced to a single stone.