Fete of Combinatorics and Computer Science
Volume 20 of the series Bolyai Society Mathematical Studies pp 6393
Solution of Peter Winkler’s Pizza Problem*†
 Josef CibulkaAffiliated withDepartment of Applied Mathematics, Charles University, Faculty of Mathematics and Physics, Malostranské nám. 25
 , Rudolf StolařAffiliated withDepartment of Applied Mathematics, Charles University, Faculty of Mathematics and Physics, Malostranské nám. 25
 , Jan KynčlAffiliated withDepartment of Applied Mathematics and Institute for Theoretical Computer Science, Charles University, Faculty of Mathematics and Physics
 , Viola MészárosAffiliated withDepartment of Applied Mathematics and Institute for Theoretical Computer Science, Charles University, Faculty of Mathematics and Physics
 , Pavel ValtrAffiliated withDepartment of Applied Mathematics and Institute for Theoretical Computer Science, Charles University, Faculty of Mathematics and Physics
 , Viola MészárosAffiliated withBolyai Institute, University of Szeged
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.
 Title
 Solution of Peter Winkler’s Pizza Problem*†
 Book Title
 Fete of Combinatorics and Computer Science
 Pages
 pp 6393
 Copyright
 2010
 DOI
 10.1007/9783642135804_4
 Print ISBN
 9783642135798
 Online ISBN
 9783642135804
 Series Title
 Bolyai Society Mathematical Studies
 Series Volume
 20
 Series ISSN
 12174696
 Publisher
 Springer Berlin Heidelberg
 Copyright Holder
 János Bolyai Mathematical Society and SpringerVerlag
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 Editors

 Gyula O. H. Katona ^{(1)}
 Alexander Schrijver ^{(2)}
 Tamás Szőnyi ^{(3)}
 Gábor Sági ^{(4)}
 Editor Affiliations

 1. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences
 2. Centre for Mathematics and Computer Science
 3. Department of Computer Science, Eötvös University
 4. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences
 Authors

 Josef Cibulka ^{(5)}
 Rudolf Stolař ^{(5)}
 Jan Kynčl ^{(6)}
 Viola Mészáros ^{(6)}
 Pavel Valtr ^{(6)}
 Viola Mészáros ^{(7)}
 Author Affiliations

 5. Department of Applied Mathematics, Charles University, Faculty of Mathematics and Physics, Malostranské nám. 25, 118 00 Praha 1, Czech Republic
 6. Department of Applied Mathematics and Institute for Theoretical Computer Science, Charles University, Faculty of Mathematics and Physics, Malostranské nám. 25, 118 00 Praha 1, Czech Republic
 7. Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, 6720 Szeged, Hungary
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