# Solution of Peter Winkler’s Pizza Problem*†

• Josef Cibulka
• Rudolf Stolař
• Jan Kynčl
• Viola Mészáros
• Pavel Valtr
• Viola Mészáros
Chapter
Part of the Bolyai Society Mathematical Studies book series (BSMS, volume 20)

## Abstract

Bob cuts a pizza into slices of not necessarily equal size and shares it with Alice by alternately taking turns. One slice is taken in each turn. The first turn is Alice’s. She may choose any of the slices. In all other turns only those slices can be chosen that have a neighbor slice already eaten. We prove a conjecture of Peter Winkler by showing that Alice has a strategy for obtaining 4/9 of the pizza. This is best possible, that is, there is a cutting and a strategy for Bob to get 5/9 of the pizza. We also give a characterization of Alice’s best possible gain depending on the number of slices. For a given cutting of the pizza, we describe a linear time algorithm that computes Alice’s strategy gaining at least 4/9 of the pizza and another algorithm that computes the optimal strategy for both players in any possible position of the game in quadratic time. We distinguish two types of turns, shifts and jumps. We prove that Alice can gain 4/9, 7/16 and 1/3 of the pizza if she is allowed to make at most two jumps, at most one jump and no jump, respectively, and the three constants are the best possible.

## Keywords

Optimal Strategy Minimum Size Characteristic Cycle Circular Sequence Optimal Turn
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. [1]
J. Cibulka, J. Kynčl, V. Mészáros, R. Stolař and P. Valtr, Generalizations of Peter Winkler’s Pizza Problem, in preparation.Google Scholar
2. [2]
H. Gourvest, LPSolve IDE v5.5, http://www.progdigy.com M. Berkelaar, J. Dirks, K. Eikland and P. Notebaert, lp_solve 5.5, http://lpsolve.sourceforge.net/5.5
3. [3]
K. Knauer, P. Micek and T. Ueckerdt, How to eat 4/9 of a pizza, manuscript.Google Scholar

© János Bolyai Mathematical Society and Springer-Verlag 2010

## Authors and Affiliations

• Josef Cibulka
• 1
• Rudolf Stolař
• 1
• Jan Kynčl
• 2
• Viola Mészáros
• 2
• Pavel Valtr
• 2
• Viola Mészáros
• 3
1. 1.Department of Applied MathematicsCharles University, Faculty of Mathematics and Physics, Malostranské nám. 25Czech Republic
2. 2.Department of Applied Mathematics and Institute for Theoretical Computer ScienceCharles University, Faculty of Mathematics and PhysicsCzech Republic
3. 3.Bolyai Institute, University of SzegedHungary