On the PSPACE-completeness of Peg Duotaire and other Peg-Jumping Games Davide Bilò Dipartimento di Scienze Umanistiche e Sociali, University of Sassari, Italy. davide.bilo@uniss.it https://orcid.org/0000-0003-3169-4300 Luciano Gualà Dipartimento di Ingegneria dell’Impresa, University of Rome “Tor Vergata”, Italy. guala@mat.uniroma2.it https://orcid.org/0000-0001-6976-5579 Stefano Leucci Institute of Theoretical Computer Science, ETH Zürich, Switzerland. stefano.leucci@inf.ethz.ch https://orcid.org/0000-0002-8848-7006 Guido Proietti Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica, University of L’Aquila, Italy, and Istituto di Analisi dei Sistemi ed Informatica, CNR, Roma, Italy. guido.proietti@univaq.it https://orcid.org/0000-0003-1009-5552 Mirko Rossi Dipartimento di Ingegneria dell’Impresa, University of Rome “Tor Vergata”, Italy. r.mirko25@gmail.com Abstract Peg Duotaire is a two-player version of the classical puzzle called Peg Solitaire. Players take turns making peg-jumping moves, and the first player which is left without available moves loses the game. Peg Duotaire has been studied from a combinatorial point of view and two versions of the game have been considered, namely the single- and the multi-hop variant. On the other hand, understanding the computational complexity of the game is explicitly mentioned as an open problem in the literature. We close this problem and prove that both versions of the game are PSPACE-complete. We also prove the PSPACE-completeness of other peg-jumping games where two players control pegs of different colors. 2012 ACM Subject Classification Theory of computation → Problems, reductions and com- pleteness Keywords and phrases peg duotaire, pspace-completeness, peg solitaire, two-player games Digital Object Identifier 10.4230/LIPIcs.FUN.2018.8 1 Introduction Peg-Jumping games are games with one or more players that are played on boards of different shapes. Each position of the board can host at most one peg, and a move consists of jumping a peg over an (horizontally or vertically) adjacent peg into an empty position. The move causes the peg that is jumped over to be removed from the board (see Figure 1). Arguably, the most popular game in this class is the single-player puzzle called Peg Solitaire (also © Davide Bilò, Luciano Gualà, Stefano Leucci, Guido Proietti, and Mirko Rossi; licensed under Creative Commons License CC-BY 9th International Conference on Fun with Algorithms (FUN 2018). Editors: Hiro Ito, Stefano Leonardi, Linda Pagli, and Giuseppe Prencipe; Article No. 8; pp. 8:1–8:15 Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany