4/9ths of a Pizza
Sometimes I get sent mathematical papers that are both interesting enough in their result and quirky enough for me to want to present them here even if the maths involved is a bit harder than the stuff I usually put it on here. I'll explain the problem and one of the basic cases, but I'll give you the link for you to read the full solution in their own words.
Alice and Bob are dividing up a pizza and each want to get as much of it as possible akin to the cake cutting problem. However the pizza is divided up into radial slices beforehand with all n pieces not necessarily of the same size. Alice starts by picking any piece and from then the two players take turns in taking pieces, but they can only pick pieces which are adjacent to the pieces which have already been picked. Here is the start of an example game from the paper:
If n is an even number then Alice can guarantee least 1/2 of the pizza. Imagine we colour in the slices red and green in an alternating fashion. There exists a colour with at least half of the pizza, say red. Alice can start with any red piece and then just take whichever red piece comes up for grabs each time. If n is odd then Alice will get to take one extra piece so it feels like Alice should do even better, but coming up with a strategy is harder than it looks.
The paper from 2008 spends 15 pages laying out the strategy which guarantees at least 4/9ths of the pizza for Alice which is the currently the best strategy we have yet found. It is unknown whether we can do any better.