International Journal of Game Theory

, Volume 47, Issue 2, pp 577–594 | Cite as

Rulesets for Beatty games

  • Lior Goldberg
  • Aviezri S. FraenkelEmail author
Original Paper


We describe a ruleset for a 2-pile subtraction game with P-positions \(\{(\lfloor \alpha n \rfloor ,\lfloor \beta n \rfloor ) : n \in \mathbb Z_{\ge 0} \}\) for any irrational \(1< \alpha < 2\), and \(\beta \) such that \(1/\alpha +1/\beta = 1\). We determine the \(\alpha \)’s for which the game can be represented as a finite modification of t-Wythoff (Holladay, Math Mag 41:7–13, 1968; Fraenkel, Am Math Mon 89(6):353–361, 1982) and describe this modification.


Subtraction games Beatty games P-positions 


  1. Beatty S (1926) Problem 3173. Am Math Mon 33:159CrossRefGoogle Scholar
  2. Cassaigne J, Duchêne E, Rigo M (2016) Nonhomogeneous Beatty sequences leading to invariant games. SIAM J Discrete Math 30:1798–1829CrossRefGoogle Scholar
  3. Coxeter HSM (1953) The golden section, phyllotaxis and Wythoff’s game. Scr Math 19:135–143Google Scholar
  4. Duchêne E, Fraenkel AS, Nowakowski RJ, Rigo M (2010) Extensions and restrictions of Wythoff’s game preserving its \({P}\)-positions. J Combin Theory Ser A 117:545–567CrossRefGoogle Scholar
  5. Duchêne E, Parreau A, Rigo M (2017) Deciding game invariance. Inf Comput 253:127–142CrossRefGoogle Scholar
  6. Duchêne E, Rigo M (2010) Invariant games. Theor Comput Sci 411(34–36):3169–3180CrossRefGoogle Scholar
  7. Erdös P, Graham RL (1980) Old and new problems and results in combinatorial number theory. Université de Genéve, L’Enseignement Mathématique, GenevaGoogle Scholar
  8. Fisher MJ, Larsson U (2011) Chromatic Nim finds a game for your solution. Cambridge University Press, Cambridge. To appear in: Larsson U (ed) Games of no chance 5, Proceedings of BIRS workshop on combinatorial games, vol 70, 2011, Banff, MSRI PublicationsGoogle Scholar
  9. Fraenkel AS (1982) How to beat your Wythoff games’ opponent on three fronts. Am Math Mon 89(6):353–361CrossRefGoogle Scholar
  10. Fraenkel AS (2013) Beating your fractional Beatty game opponent and: what’s the question to your answer? In: Advances in combinatorics, vol 63. Springer, Berlin. In Memory of Herbert S. Wilf, Proceedings of Waterloo workshop on computer algebra, 2011, IS Kotsireas and EV Zima (eds)Google Scholar
  11. Fraenkel AS (2015) The rat game and the mouse game. In: Nowakowski RJ (ed) Games of no chance 4, Proceedings of BIRS workshop on combinatorial games, 2008, Banff, vol 63. MSRI Publ., Cambridge University Press, CambridgeGoogle Scholar
  12. Fraenkel AS, Larsson U (2011) Take-away games and the notion of \(k\)-invariance. Cambridge University Press, Cambridge. To appear in: Larsson U (ed) Games of no chance 5, Proceedings of BIRS workshop on combinatorial games, vol 70, 2011, Banff. MSRI PublicationsGoogle Scholar
  13. Fraenkel AS, Larsson U (2017) Games on arbitrarily large rats and playability (preprint) Google Scholar
  14. Goldberg L, Fraenkel AS (2013)Extensions of Wythoff’s game (preprint) Google Scholar
  15. Golomb SW (1966) A mathematical investigation of games of “take-away”. J Combin Theory 1(4):443–458CrossRefGoogle Scholar
  16. Hardy GH, Wright EM (2008) An introduction to the theory of numbers, 6th edn. Oxford University Press, OxfordGoogle Scholar
  17. Ho NB (2012) Two variants of Wythoff’s game preserving its \({P}\)-positions. J Combin Theory Ser A 119:1302–1314CrossRefGoogle Scholar
  18. Holladay JC (1968) Some generalizations of Wythoff’s game and other related games. Math Mag 41:7–13CrossRefGoogle Scholar
  19. Larsson U (2012) The \(\star \)-operator and invariant subtraction games. Theor Comput Sci 422:52–58CrossRefGoogle Scholar
  20. Larsson U, Hegarty P, Fraenkel AS (2011) Invariant and dual subtraction games resolving the Duchêne–Rigo conjecture. Theor Comput Sci 412(8–10):729–735CrossRefGoogle Scholar
  21. Larsson U, Weimerskirch M (2013) Impartial games whose rulesets produce given continued fractions (arXiv preprint). arXiv:1302.0271
  22. Wythoff WA (1907) A modification of the game of Nim. Nieuw Arch Wiskd 7:199–202Google Scholar
  23. Yaglom AM, Yaglom IM (1967) Challenging mathematical problems with elementary solutions., vol II. Holden-Day, San Francisco (translated by J. McCawley, Jr., revised and edited by B. Gordon) Google Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of Computer Science and Applied MathematicsWeizmann Institute of ScienceRehovotIsrael

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