International Journal of Game Theory

, Volume 47, Issue 2, pp 613–652

# Wythoff partizan subtraction

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

## 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

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.

## References

1. Albert MH, Nowakowski RJ, Wolfe D (2007) Lessons in play. A K Peters Ltd, NatickGoogle Scholar
2. Austin RB (1976) Impartial and partizan games, M.Sc. Thesis. The University of CalgaryGoogle Scholar
3. Beatty S (1926) Problem 3173. Am Math Mon 33:159
4. Berlekamp ER, Conway JH, Guy RK (2001–2004) Winning ways for your mathematical plays, 2nd edn, vols. 1–4. A K Peters Ltd, NatickGoogle Scholar
5. Berstel J, Séébold P (2002) Finite and infinite words. In: Lothaire M (eds) Algebraic combinatorics on words, chapter 2. Cambridge University Press, Cambridge (available online for free download)Google Scholar
6. Calistrate D (1996) The reduced canonical form of a game, games of no chance. Cambridge University Press, Cambridge, pp 409–416Google Scholar
7. Conway JH (2001) On numbers and games, 2nd edn. A K Peters Ltd, NatickGoogle Scholar
8. Duchene E, Fraenkel AS, Gurvich V, Ho NB, Kimberling C, Larsson U (2018) Wythoff visons. In: Games of no chance 5. Cambridge University Press, Cambridge (to appear) Google Scholar
9. Fraenkel AS, Kotzig A (1987) Partizan octal games: partizan subtraction games. Int J Game Theory 16:145–154
10. Grossman JP, Nowakowski RJ (2015) A ruler regularity in hexadecimal games, games of no chance 4. Cambridge University Press, Cambridge, pp 115–128Google Scholar
11. Grossman JP, Siegel A (2009) Reductions of partizan games, games of no chance 3. Cambridge University Press, Cambridge, pp 437–456Google Scholar
12. Kimberling C (2008) Complementary equations and Wythoff sequences. J Integer Seq 11:Article 0.8.33Google Scholar
13. Larsson U, Hegarty P, Fraenkel AS (2011) Invariant and dual subtraction games resolving the Duchêne–Rigo conjecture. Theor Comput Sci 412:729–735
14. McKay NA, Nowakowski RJ, Siegel AA (2015) Navigating the maze, games of no chance 4. Cambridge University Press, Cambridge, pp 183–194Google Scholar
15. Mesdal GA (2009) Partizan splittles, games of no chance 3. Cambridge University Press, Cambridge, pp 457–471Google Scholar
16. Nowakowski RJ, Ottaway P (2011) Option-closed games. Contrib Discrete Math 6:142–153Google Scholar
17. Plambeck T (1995) Partisan subtraction games working notes. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.391.7386
18. Siegel AA (2005) Finite excluded subtraction sets and infinite modular Nim, M.Sc. Thesis. Dalhousie UniversityGoogle Scholar
19. Siegel AN (2013) Combinatorial game theory. American Mathematical Society, Providence
20. Silber R (1976) A Fibonacci property of Wythoff pairs. Fibonacci Q 14(4):380–384Google Scholar
21. Wythoff WA (1907) A modification of the game of Nim. Nieuw Arch Wisk 7:199–202Google Scholar

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