In the past decade there's been a fair bit of attention paid to the "pizza theorem" and its generalizations. The basic pizza theorem says that if you choose an arbitrary point P in the interior of a disk, draw 4 lines through P that meet at 45 degree angles (by which I mean "45 degrees and multiples thereof", for those of a pedantic turn of mind), and color the 8 resulting wedges alternately black and white, the total area of the black wedges equals the total area of the white edges. I mentioned the result to Pete Winkler, and he quickly came up with an approach that allows one to give a purely mental proof of the result. I'm wondering if the proof is new. I'll give people a chance to come up with it on their own before giving hints or giving details. Of course, reasonable people may differ about what counts as an argument one can construct "in one's head". In both cases, there was an algebraic calculation that needed to be done, which I was able to do in my head with some slight effort. In the course of finding the proof, I inadvertently proved a slightly different result as well, in which the pizza is divided into 8 pieces in a different fashion, namely, by joining the (arbitrary internal) point P to 8 equally-spaced points on the boundary of the disk. In both cases, the method of proof automatically establishes a stronger result, in which you divide the pizza into 4n pieces and divide the pieces among n people in cyclic fashion, so that each of the n people gets 4 slices whose areas turn out to sum to 1/n times the area of the pizza. I'm hoping that people who know the pizza-problem literature will chime in at some point, after the non-initiates have had a chance to mull over the problem on their own over the next couple of days. Jim Propp