Large Peg-Army Maneuvers

TL;DR

This paper investigates algorithmic problems related to peg solitaire, proving NP-completeness for reachability and solving open questions in deploying peg armies, highlighting the game's computational complexity and strategic depth.

Contribution

It establishes NP-completeness for Solitaire-Reachability and solves open problems in Solitaire-Army using relaxed constraints, advancing understanding of the game's computational aspects.

Findings

Solitaire-Reachability is NP-complete.

Solved open questions in Solitaire-Army.

Provided visual animations of solutions.

Abstract

Despite its long history, the classical game of peg solitaire continues to attract the attention of the scientific community. In this paper, we consider two problems with an algorithmic flavour which are related with this game, namely Solitaire-Reachability and Solitaire-Army. In the first one, we show that deciding whether there is a sequence of jumps which allows a given initial configuration of pegs to reach a target position is NP-complete. Regarding Solitaire-Army, the aim is to successfully deploy an army of pegs in a given region of the board in order to reach a target position. By solving an auxiliary problem with relaxed constraints, we are able to answer some open questions raised by Cs\'ak\'any and Juh\'asz (Mathematics Magazine, 2000). To appreciate the combinatorial beauty of our solutions, we recommend to visit the gallery of animations provided at http://solitairearmy.isnphard.com.