International Journal of Game Theory

, Volume 47, Issue 2, pp 613–652 | Cite as

Wythoff partizan subtraction

  • Urban LarssonEmail author
  • Neil A. McKay
  • Richard J. Nowakowski
  • Angela A. Siegel
Original Paper


We introduce a class of normal-play partizan games, called Complementary Subtraction. These games are instances of Partizan Subtraction where we take any set A of positive integers to be Left’s subtraction set and let its complement be Right’s subtraction set. In wythoff partizan subtraction we take the set A and its complement B from wythoff nim, as the two subtraction sets. As a function of the heap size, the maximum size of the canonical forms grows quickly. However, the value of the heap is either a number or, in reduced canonical form, a switch. We find the switches by using properties of the Fibonacci word and standard Fibonacci representations of integers. Moreover, these switches are invariant under shifts by certain Fibonacci numbers. The values that are numbers, however, are distinct, and we can find their binary representation in polynomial time using a representation of integers as sums of Fibonacci numbers, known as the ternary (or “the even”) Fibonacci representation.


Combinatorial game theory Complementary subtraction Fibonacci sequence Partizan subtraction game Reduced canonical form Sturmian word Wythoff’s sequences 

Mathematics Subject Classification

91A46 11B39 



We dedicate this work to Urban’s father Göran Larsson. Without his support, and love for mathematics, this project would not have been possible. Thanks also to the referees for their helpful comments on the presentation.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsDalhousie UniversityHalifaxCanada
  2. 2.Robert Gordon UniversityAberdeenUK

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