Solitaire Clobber: A New Game of Strategy and Reduction

Title: Solitaire Clobber: A New Game of Strategy and Reduction

Research Question: How can one reduce the number of stones in a Solitaire Clobber game, and what are the implications for the game's strategy and complexity?

Methodology: The researchers studied the game of Solitaire Clobber, a two-player board game with complete information. They focused on 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. They analyzed the reducibility of different board configurations, including one-dimensional (single row) and two-dimensional (rectangular checkerboard) arrangements.

Results: The researchers found that the number of stones in a one-dimensional Solitaire Clobber game can be reduced to approximately n/4 stones, with an additional stone added if n is divisible by 3. They also showed that in two-dimensional games with at least two rows and two columns, a checkerboard configuration can be reduced to a single stone if n is not a multiple of 3, and two stones otherwise. Furthermore, they proved that deciding whether an arbitrary board configuration is 1-reducible (i.e., can be reduced to a single stone) is NP-complete.

Implications: These results have significant implications for the game's strategy and complexity. The ability to reduce the number of stones in a one-dimensional game to approximately n/4 stones suggests that players should focus on achieving this optimal configuration. The NP-completeness of deciding 1-reducibility in two-dimensional games means that finding the optimal move in these cases can be computationally challenging, potentially leading to a draw or a stalemate situation. This also highlights the importance of understanding the reducibility of different board configurations for developing effective strategies in Solitaire Clobber.

Link to Article: https://arxiv.org/abs/0204017v1 Authors: arXiv ID: 0204017v1