This classification has now been found. It's not the most ground-breaking research. Deierman and Mabry won't become household names, and won't be nominated for a Fields Medal for this work. But I love this sort of thing because it shows how the application of some intelligence, hard work, and cleverness (not the same thing as intelligence) can simplify decidedly nontrivial problems.
The solution, btw, is simple enough to write down here:
- If at least one cut passes through the center, then the two diners will always get equal amounts.
- If you cut your pizza an even number of times (say, 4), then the two diners will always get equal amounts.
- If you cut your pizza an odd number of times and that odd number is expressible as 4n+3 (say, 3, 7, etc.) then the person who eats the piece that contains the center of the pizza gets more.
- If you cut your pizza an odd number of times and that odd number is expressible as 4n+1 (say, 5, 9, etc.) then the person who eats the piece that does not contain the center of the pizza gets more.
One cool consequence of this result is that, if you have a pizza with an odd number of unequal cuts, and you want to share it equally between two people, all you have to do is just cut it one more time (the new cut must pass through the common intersection of all the previous cuts). Then as long as the diners take alternate pieces, they will get equal amounts. That's a nontrivial, counterintuitive result, and with a practical application!