Skip to main content

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

Buy Article:

$12.00 plus tax (Refund Policy)

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.
No Reference information available - sign in for access.
No Citation information available - sign in for access.
No Supplementary Data.
No Data/Media
No Metrics

Document Type: Research Article

Publication date: 2009-05-01

More about this publication?
  • Access Key
  • Free content
  • Partial Free content
  • New content
  • Open access content
  • Partial Open access content
  • Subscribed content
  • Partial Subscribed content
  • Free trial content
Cookie Policy
X
Cookie Policy
Ingenta Connect website makes use of cookies so as to keep track of data that you have filled in. I am Happy with this Find out more