International Journal of Game Theory

, Volume 47, Issue 2, pp 557–576 | Cite as

Sterling stirling play

  • Michael Fisher
  • Richard J. Nowakowski
  • Carlos SantosEmail author
Original Paper


In this paper we analyze a recently proposed impartial combinatorial ruleset that is played on a permutation of the set \(\left[ n\right] \). We call this ruleset Stirling Shave. A procedure utilizing the ordinal sum operation is given to determine the nim value of a given normal play position. Additionally, we enumerate the number of permutations of \(\left[ n\right] \) which are \(\mathcal {P}\)-positions. The formula given involves the Stirling numbers of the first-kind. We also give a complete analysis of the Misère version of Stirling Shave using Conway’s genus theory. An interesting by-product of this analysis is insight into how the ordinal sum operation behaves in Misère Play.


Combinatorial game theory Impartial games Normal Play Misère Play Ordinal sum Stirling numbers of the first kind 


  1. Albert M, Nowakowski RJ, Wolfe D (2007) Lessons in play: an introduction to combinatorial game theory. A. K. PetersGoogle Scholar
  2. Berlekamp ER, Conway JH, Guy RK (1982) Winning ways. Academic Press, LondonGoogle Scholar
  3. Bernstein M, Sloane N (1995) Some canonical sequences of integers. Linear Algebra Appl 226(228):57–72CrossRefGoogle Scholar
  4. Bouton CL (1902) Nim, a game with a complete mathematical theory. Ann Math 3(2):35–39Google Scholar
  5. Brown JI, Cox D, Hoefel A, McKay N, Milley R, Nowakowski RJ, Siegel AA (2017) Polynomial profiles of placement games. Games of no chance 5. (to appear)Google Scholar
  6. Conway JH (1976) On numbers and games. Academic PressGoogle Scholar
  7. Farr GE (2003) The Go polynomials of a graph. Theor Comp Sci 306:1–18CrossRefGoogle Scholar
  8. Grundy PM (1939) Mathematics and games. Eureka Google Scholar
  9. Hetyei G (2009) Enumeration by kernel positions. Adv Appl Math 42(4):445–470CrossRefGoogle Scholar
  10. Hetyei G (2010) Enumeration by kernel positions for strongly Bernoulli type truncation games on words. J Combin Theory Ser A 117:1107–1126CrossRefGoogle Scholar
  11. McKay NA (2016) Forms and values of number-like and nimber-like games. Dalhousie University, Ph.DGoogle Scholar
  12. McKay NA, Milley R, Nowakowski RJ (2015) Misère-play Hackenbush Sprigs. Int J Game Theory 45:1–12Google Scholar
  13. Nowakowski RJ, Ottaway P (2011) Option-closed games. Contrib Disc Math 6:142–153Google Scholar
  14. Nowakowski RJ, Renault G, Lamoureux E, Mellon S, Miller T (2013) The game of Timber!. J Combin Math Combin Comput 85:213–225Google Scholar
  15. OEIS Foundation Inc. (2011) The on-line encyclopedia of integer sequences. A089064
  16. Siegel AA (2011) On the structure of games and their posets. Dalhousie University, Ph.DGoogle Scholar
  17. Siegel AN (2013) Combinatorial game theory. American Math. SocGoogle Scholar
  18. Sprague RP (1935) Über mathematische Kampfspiele. Tohoku Math JGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.West Chester UniversityWest ChesterUSA
  2. 2.Dalhousie UniversityHalifaxCanada
  3. 3.Center for Functional Analysis, Linear Structures and ApplicationsLisbonPortugal

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