Three-player nim with podium rule

  • Richard J. Nowakowski
  • Carlos P. SantosEmail author
  • Alexandre M. Silva
Original Paper


If a combinatorial game involves more than two players, the problem of coalitions arises. To avoid the problem, Shuo-Yen Robert Li analyzed three-player nim with the podium rule, that is, if a player cannot be last, he should try to be last but one. With that simplification, he proved that a disjunctive sum of nim piles is a \({\mathcal {P}}\)-position if and only if the sum modulo 3 of the binary representations of the piles is equal to zero. In this paper, we extend the result in order to understand the complete characterization of the outcome classes, the possible reductions of the game forms, the equivalence classes under the equality of games and related canonical forms.


Combinatorial game theory Impartial games nim Three-player games Podium rule 



The authors are grateful to the anonymous referee for his high quality report. That has greatly improved this paper.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2020

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsDalhousie UniversityHalifaxCanada
  2. 2.Center for Functional Analysis Linear Structures and ApplicationsISEL-IPL, University of LisbonLisbonPortugal
  3. 3.University of MinhoBragaPortugal

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