## Abstract

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.

## Keywords

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

91A46 11B39## Notes

### Acknowledgements

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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