International Journal of Game Theory

, Volume 47, Issue 2, pp 595–611 | Cite as

Global Fibonacci nim

  • Urban LarssonEmail author
  • Simon Rubinstein-Salzedo
Original Paper


Fibonacci nim is a popular impartial combinatorial game, usually played with a single pile of stones: two players alternate in removing no more than twice the previous player’s removal. The game is appealing due to its surprising connections with the Fibonacci numbers and the Zeckendorf representation. In this article, we investigate some properties of a variant played with multiple piles of stones, and solve the 2-pile case. A player chooses one of the piles and plays as in Fibonacci nim, but here the move-size restriction is a global parameter, valid for any pile.


Combinatorial game Complementary value Complementary equation Fibonacci sequence Fibonacci word Impartial game Power-of-two nim Sturmian word Zeckendorf representation 



Part of the work for this paper was completed at the Games at Dal workshop at Dalhousie University in Halifax, Nova Scotia, in August 2015. The first author was supported by the Killam Trusts. We would like to thank the referees for their helpful suggestions.


  1. Albert M, Nowakowski R, Wolfe D (2007) Lessons in play: an introduction to combinatorial game theory. CRC Press, Boca RatonGoogle Scholar
  2. Berstel J (1986) Fibonacci words—a survey. In: Rozenberg G, Salomaa A (eds) The book of L. Springer, Berlin, Heidelberg, pp 13–27Google Scholar
  3. Grundy PM (1939) Mathematics and games. Eureka 2(6–8):21Google Scholar
  4. Kimberling C (2008) Complementary equations and Wythoff sequences. J Integer Seq 11:3. Article 08.3.3, 8Google Scholar
  5. Knuth DE (1997) Fundamental algorithms: the art of computer programming, vol 1, 3rd edn. Addison-Wesley, ReadingGoogle Scholar
  6. Larsson U (2009) 2-pile nim with a restricted number of move-size imitations. Integers 9(6):671–690CrossRefGoogle Scholar
  7. Lekkerkerker CG (1952) Representation of natural numbers as a sum of Fibonacci numbers. Simon Stevin 29:190–195Google Scholar
  8. Lothaire M (2002) Algebraic combinatorics on words, vol 90 of encyclopedia of mathematics and its applications. Cambridge University Press, Cambridge. A collective work by Jean Berstel, Dominique Perrin, Patrice Seebold, Julien Cassaigne, Aldo De Luca, Steffano Varricchio, Alain Lascoux, Bernard Leclerc, Jean-Yves Thibon, Veronique Bruyere, Christiane Frougny, Filippo Mignosi, Antonio Restivo, Christophe Reutenauer, Dominique Foata, Guo-Niu Han, Jacques Desarmenien, Volker Diekert, Tero Harju, Juhani Karhumaki and Wojciech Plandowski, With a preface by Berstel and PerrinGoogle Scholar
  9. Larsson U, Rubinstein-Salzedo S (2016) Grundy values of Fibonacci nim. Int J Game Theory 45(3):617–625CrossRefGoogle Scholar
  10. Sprague R (1935) Über mathematische Kampfspiele. Tôhoku Math J 41:438–444Google Scholar
  11. Whinihan MJ (1963) Fibonacci nim. Fibonacci Q 1(4):9–13Google Scholar
  12. Zeckendorf E (1972) Représentation des nombres naturels par une somme de nombres de Fibonacci ou de nombres de Lucas. Bull Soc R Sci Liège 41:179–182Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.The Faculty of Industrial Engineering and ManagementTechnion–Israel Institute of TechnologyHaifaIsrael
  2. 2.Euler CirclePalo AltoUSA

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