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

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Two friends, Gray and White, dine out to share a pizza. Their server brings a pizza that has been sliced using N straight, concurrent, equiangular cuts. This should enable Gray and White to share the pizza equally, presuming they both take the same number (N) of slices. The trouble is, the pizza-cutter missed the mark—the point of concurrency is not the center! Questions now arise. In particular, How can the pizza be equally shared? And, Who says they want to share equally?

It has been known since the 1960s that when N is even and greater than 2, an answer to the first question is for Gray and White to choose alternate slices about the point P of concurrency. (The figures below show the situation for N=6 and N=7.) This alternation scheme also results in equal shares for any N, quite obviously, if the center O lies on one of the cuts. But if N is odd, and if O does not lie on a cut, then, as has been known since the 1990s, this method of alternating slices does not result in equal sharing. Those more inclined to the second question above now want to know, Who gets the most pizza?

It was conjectured by Stan Wagon and others, that for N=3,7,11,15,…, whoever gets the center gets the most pizza, while for N=5,9,13,17,…, whoever gets the center gets the least. We prove this Pizza Conjecture by first showing its equivalence to a (pretty wild) trigonometric inequality. This inequality is proved with the aid of a theorem that counts lattice paths. Our main theorem is sufficiently general that, as a bonus, results concerning the equiangular slicing of other dishes are obtained.
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Document Type: Research Article

Publication date: 01 May 2009

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