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

Theorem 1 (Cheese Pizza Theorem). For a positive integer N, divide a pizza into 2N slices by choosing an arbitrary point P in the pizza and making N straight cuts through P, the cuts meeting to form 2N equal angles. Alternately color adjacent slices gray and white. Let O denote the center of the pizza. The total areas of gray and white will then satisfy the following. (a) When N ≥ 4 is even, the gray area equals the white, but for all other N, gray equals white if and only if O lies on a cut. (b) If O does not lie on a cut and N = 1, N = 2, or N is odd with N ≡ 3 (mod 4), then gray exceeds white if and only if O lies in a gray slice. (c) If O does not lie on a cut and N is odd with N ≥ 5 and N ≡ 1 (mod 4), then white exceeds gray if and only if O lies in a gray slice.