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Fete of Combinatorics and Computer Science

Volume 20 of the series Bolyai Society Mathematical Studies pp 63-93

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

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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.