Of Cheese and Crust: A Proof of the Pizza Conjecture and Other Tasty Results

Document Type : Translation Paper

Author

Faculty of Mathematics, K. N. Toosi University of Technology Tehran, Iran

Abstract

Translator's abstract: For a given natural number $N$, a circular pizza is divided into $2N$ equiangular slices by means of $N$ straight, concurrent cuts at an arbitrary point $P$ of the interior of the pizza. Then, each slice of the pizza is shared by two individuals (``Gray" and ``White"), who alternate slices. A natural question that may arise is that does the area of the gray slices exceed that of white slices? When $N$ is even, an answer to this question has already been given. In this paper, whenever $N$ is an odd number, a complete answer to this problem is given. Moreover, the method of proof are generalized to three dimensional pizzas, so called ``calzones", that are some type of pizzas which are obtained by filling the space above the circular pizzas with cheese surrounded by means of upper surfaces such as paraboloid, semi-ellipsoid or cone, and both volumes and surface areas of the Gray and White slices are computed.

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References

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Mathematics and Society
Volume 10, Issue 4 - Serial Number 4
December 2025
Pages 61-84

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History

  • Receive Date: 16 January 2025
  • Revise Date: 02 June 2025
  • Accept Date: 13 April 2025
  • Publish Date: 20 February 2026

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