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Completing the Complexity Classification of 2-Solo Chess: Knights and Kings Are Hard

Authors Kolja Kühn , Wendy Yi

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LIPIcs.FUN.2026.27.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Solo chess
  • puzzle games
  • board games
  • NP-completeness

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Abstract

We extend the study of the 2-Solo Chess problem which was first introduced by Aravind, Misra, and Mittal in 2022. 2-Solo Chess is a single-player variant of chess in which the player must clear the board via captures such that only one piece remains, with each piece capturing at most twice.
It is known that the problem is solvable in polynomial time for instances containing only pawns, while it becomes NP-complete for instances restricted to rooks, bishops, or queens. In this work, we complete the complexity classification by proving that 2-Solo Chess is NP-complete if the instance contains only knights or only kings.

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Kolja Kühn and Wendy Yi. Completing the Complexity Classification of 2-Solo Chess: Knights and Kings Are Hard. In 13th International Conference on Fun with Algorithms (FUN 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 366, pp. 27:1-27:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.FUN.2026.27

Author Details

Kolja Kühn
  • Institute of Theoretical Informatics, Karlsruhe Institute of Technology, Germany
Wendy Yi
  • Institute of Theoretical Informatics, Karlsruhe Institute of Technology, Germany

References

  1. N. R. Aravind, Neeldhara Misra, and Harshil Mittal. Chess is hard even for a single player. Theor. Comput. Sci., 1015:114726, 2024. URL: https://doi.org/10.1016/j.tcs.2024.114726.
  2. N.R. Aravind, Neeldhara Misra, and Harshil Mittal. Chess Is Hard Even for a Single Player. In Pierre Fraigniaud and Yushi Uno, editors, 11th International Conference on Fun with Algorithms, volume 226 of Leibniz International Proceedings in Informatics (LIPIcs), pages 5:1-5:20, 2022. URL: https://doi.org/10.4230/LIPIcs.FUN.2022.5.
  3. Davide Bilò, Luca Di Donato, Luciano Gualà, and Stefano Leucci. Uniform-Budget Solo Chess with Only Rooks or Only Knights Is Hard. In Andrei Z. Broder and Tami Tamir, editors, 12th International Conference on Fun with Algorithms, volume 291 of Leibniz International Proceedings in Informatics (LIPIcs), pages 4:1-4:19, 2024. URL: https://doi.org/10.4230/LIPIcs.FUN.2024.4.
  4. Davide Bilò, Luca Di Donato, Luciano Gualà, and Stefano Leucci. Uniform-budget solo chess with only rooks or only knights is hard. Theor. Comput. Sci., 1056:115544, 2025. URL: https://doi.org/10.1016/J.TCS.2025.115544.
  5. Josh Brunner, Lily Chung, Michael Coulombe, Erik D. Demaine, Timothy Gomez, and Jayson Lynch. Complexity of solo chess with unlimited moves. In Jin Akiyama, Hiro Ito, and Toshinori Sakai, editors, Discrete and Computational Geometry, Graphs, and Games, pages 149-174. Springer Nature Switzerland, 2026. URL: https://doi.org/10.1007/978-3-032-00281-5_10.
  6. Chess.com. Solo chess. URL: https://www.chess.com/solo-chess.
  7. Kolja Kühn. Complexity of Solo Chess Variants. Master’s thesis, Karlsruhe Institute of Technology, 2024. URL: https://scale.iti.kit.edu/_media/resources/theses/ma_kolja_kuehn.pdf.
  8. Craig A. Tovey. A simplified NP-complete satisfiability problem. Discrete Applied Mathematics, 8(1):85-89, 1984. URL: https://doi.org/10.1016/0166-218X(84)90081-7.
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