Document Open Access Logo

Nimber-Preserving Reduction: Game Secrets And Homomorphic Sprague-Grundy Theorem

Authors Kyle W. Burke, Matthew Ferland, Shang-Hua Teng

Thumbnail PDF


  • Filesize: 0.73 MB
  • 17 pages

Document Identifiers

Author Details

Kyle W. Burke
  • Department of Computer Science, Plymouth State University, NH, USA
Matthew Ferland
  • Department of Computer Science, University of Southern California, Los Angeles, CA, USA
Shang-Hua Teng
  • Department of Computer Science, University of Southern California, Los Angeles, CA, USA

Cite AsGet BibTex

Kyle W. Burke, Matthew Ferland, and Shang-Hua Teng. Nimber-Preserving Reduction: Game Secrets And Homomorphic Sprague-Grundy Theorem. In 11th International Conference on Fun with Algorithms (FUN 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 226, pp. 10:1-10:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


The concept of nimbers - a.k.a. Grundy-values or nim-values - is fundamental to combinatorial game theory. Beyond the winnability, nimbers provide a complete characterization of strategic interactions among impartial games in disjunctive sums. In this paper, we consider nimber-preserving reductions among impartial games, which enhance the winnability-preserving reductions in traditional computational characterizations of combinatorial games. We prove that Generalized Geography is complete for the natural class, ℐ^P, of polynomially-short impartial rulesets, under polynomial-time nimber-preserving reductions. We refer to this notion of completeness as Sprague-Grundy-completeness. In contrast, we also show that not every PSPACE-complete ruleset in ℐ^P is Sprague-Grundy-complete for ℐ^P. By viewing every impartial game as an encoding of its nimber - a succinct game secret richer than its winnability alone - our technical result establishes the following striking cryptography-inspired homomorphic theorem: Despite the PSPACE-completeness of nimber computation for ℐ^P, there exists a polynomial-time algorithm to construct, for any pair of games G₁, G₂ in ℐ^P, a Generalized Geography game G satisfying: nimber(G) = nimber(G₁) ⊕ nimber(G₂).

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
  • Combinatorial Games
  • Nim
  • Generalized Geography
  • Sprague-Grundy Theory
  • Grundy value
  • Computational Complexity
  • Functional-Preserving Reductions


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Selim G Akl. On the importance of being quantum. Parallel processing letters, 20(03):275-286, 2010. Google Scholar
  2. D. Applegate, G. Jacobson, and D. Sleator. Computer analysis of Sprouts. Technical Report CMU-CS-91-144, Carnegie Mellon University Computer Science, 1991. Google Scholar
  3. Gabriel Beaulieu, Kyle G. Burke, and Éric Duchêne. Impartial coloring games. Theoret. Comput. Sci., 485:49-60, 2013. URL:
  4. Elwyn R. Berlekamp, John H. Conway, and Richard K. Guy. Winning Ways for your Mathematical Plays, volume 1. A K Peters, Wellesley, Massachsetts, 2001. Google Scholar
  5. Charles L. Bouton. Nim, a game with a complete mathematical theory. Annals of Mathematics, 3(1/4):pp. 35-39, 1901. Google Scholar
  6. K. Burke and O. George. A PSPACE-complete graph nim. In Urban Larsson, editor, Games of No Chance 5, volume 70 of Mathematical Sciences Research Institute Publications, pages 259-269. Cambridge University Press, 2019. Google Scholar
  7. Kyle Burke, Matthew Ferland, and Shang-Hua Teng. Quantum combinatorial games: Structures and computational complexity. CoRR, abs/2011.03704, 2020. URL:
  8. Kyle Burke, Matthew Ferland, and Shang-Hua Teng. Transverse Wave: An impartial color-propagation game inspried by social influence and quantum nim. Integers, 21B:A3, 30, 2021. Google Scholar
  9. Kyle Burke, Matthew Ferland, and Shang-Hua Teng. Winning the war by (strategically) losing battles: Settling the complexity of Grundy-values in undirected geography. In Proceedings of the 62nd Annual Symposium on Foundations of Computer Science (FOCS). IEEE, 2021. Google Scholar
  10. Kyle W. Burke and Shang-Hua Teng. Atropos: A PSPACE-complete Sperner triangle game. Internet Mathematics, 5(4):477-492, 2008. URL:
  11. John H. Conway. On Numbers and Games (2. ed.). A K Peters, 2000. Google Scholar
  12. Erik D Demaine, Martin L Demaine, Nicholas JA Harvey, Ryuhei Uehara, Takeaki Uno, and Yushi Uno. UNO is hard, even for a single player. Theoretical Computer Science, 521:51-61, 2014. Google Scholar
  13. Paul Dorbec and Mehdi Mhalla. Toward quantum combinatorial games. arXiv preprint arXiv:1701.02193, 2017. Google Scholar
  14. S. Even and R. E. Tarjan. A combinatorial problem which is complete in polynomial space. J. ACM, 23(4):710-719, October 1976. Google Scholar
  15. Aviezri S Fraenkel. Complexity, appeal and challenges of combinatorial games. Theoretical Computer Science, 313(3):393-415, 2004. Google Scholar
  16. Aviezri S. Fraenkel and David Lichtenstein. Computing a perfect strategy for n x n chess requires time exponential in n. J. Comb. Theory, Ser. A, 31(2):199-214, 1981. URL:
  17. Aviezri S. Fraenkel, Edward R. Scheinerman, and Daniel Ullman. Undirected edge geography. Theor. Comput. Sci., 112(2):371-381, 1993. Google Scholar
  18. Masahiko Fukuyama. A nim game played on graphs. Theor. Comput. Sci., 1-3(304):387-399, 2003. URL:
  19. David Gale. The game of Hex and the Brouwer fixed-point theorem. American Mathematical Monthly, 10:818-827, 1979. Google Scholar
  20. Adam Glos and Jarosław Adam Miszczak. The role of quantum correlations in cop and robber game. Quantum Studies: Mathematics and Foundations, 6(1):15-26, 2019. Google Scholar
  21. Allan Goff. Quantum tic-tac-toe: A teaching metaphor for superposition in quantum mechanics. American Journal of Physics, 74(11):962-973, 2006. Google Scholar
  22. Daniel Grier. Deciding the winner of an arbitrary finite poset game is PSPACE-complete. In Proceedings of the 40th International Conference on Automata, Languages, and Programming - Volume Part I, ICALP'13, pages 497-503, Berlin, Heidelberg, 2013. Springer-Verlag. Google Scholar
  23. P. M. Grundy. Mathematics and games. Eureka, 2:198 - -211, 1939. Google Scholar
  24. David Lichtenstein and Michael Sipser. Go is polynomial-space hard. J. ACM, 27(2):393-401, 1980. Google Scholar
  25. John F. Nash. Some Games and Machines for Playing Them. RAND Corporation, Santa Monica, CA, 1952. Google Scholar
  26. C. H. Papadimitriou. Computational Complexity. Addison Wesley, Reading, Massachsetts, 1994. Google Scholar
  27. S. Reisch. Hex ist PSPACE-vollständig. Acta Inf., 15:167-191, 1981. Google Scholar
  28. Thomas J. Schaefer. On the complexity of some two-person perfect-information games. Journal of Computer and System Sciences, 16(2):185-225, 1978. Google Scholar
  29. Aaron N. Siegel. On the structure of misère impartial games, 2021. URL:
  30. Michael Soltys and Craig Wilson. On the complexity of computing winning strategies for finite poset games. Theory Comput. Syst., 48:680-692, April 2011. Google Scholar
  31. R. P. Sprague. Über mathematische Kampfspiele. Tôhoku Mathematical Journal, 41:438 - -444, 1935-36. Google Scholar
  32. David Wolfe. Go endgames are PSPACE-hard. In Richard J. Nowakowski, editor, More Games of No Chance, volume 42 of Mathematical Sciences Research Institute Publications, pages 125-136. Cambridge University Press, 2002. Google Scholar
  33. D. Zeilberger. Chomp, recurrences and chaos. Journal of Difference Equations and Applications, 10:1281-1293, 2004. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail