How to Fairly Cut a Cake, According to Math

How the I cut, you choose method works for more than two people

If you’ve ever split food with a friend, you’ve probably used the I cut, you choose method where one person splits the food — cake, pizza, or whatever else it may be — into what he or she believes to be two even pieces and you get to choose whichever one you like. This is a fair division method because you each get to enjoy eating what you think is at least half the food. Your friend is happy because he or she thinks the pieces were divided evenly and you are happy because you chose whichever one you considered biggest (which is logically at least half). However, if you want to share your food with a whole group of friends, the problem gets a fair bit more complicated. Who does the cutting? Who gets to choose first?

Over the years, mathematicians have suggested many methods and solutions to this dilemma, but here are two of the simplest. They both assume that the cake in question is rectangular, but can be modified for any shape.

Last Diminisher Method

This algorithm was first suggested by Stefan Banach and Bronislaw Knaster, two Polish mathematicians. Let’s consider it for three people (Alfred, Bob and Claire). Alfred looks at the cake first. He makes a cut that he thinks leaves one third of the cake to the right of his knife. Bob then looks at the cake. If he thinks that more than a third of the cake is to the right of the knife, he cuts off whatever piece he thinks leaves a third. Claire gets to do the same.

The trick is that the last person to make a cut gets to keep that piece of cake. If both Bob and Claire thought that Alfred made the right cut to start with, he gets the first piece. The same can be said for Bob if he chooses to make a cut and Claire thinks it was the right one. There is no advantage to leaving more than a third because someone else will cut off a tiny amount and get to keep that piece of cake. However, there is no advantage to cutting a piece too small because then nobody else will want it and you will be stuck with it.

Once only two people are remaining, they can use the I cut, you choose method to split it fairly. It can be generalised for n number of people where each person aims for 1/n of the cake.

This algorithm is fairly simple and should leave everyone happy, but may result in a cake with many cuts in it. It will still taste delicious, but may not be as aesthetically pleasing.

Moving Knife Algorithm

This method is even simpler, although it requires a group of friends that are able to visually estimate a bit better. Choose one person to be the cutter. That person starts with a knife on the far left and slowly moves it to the right. At any point, any person can call “cut!” The cutter then cuts the piece and hands it to the person who called out.

Nobody will make the call before the piece gets to be what they think is equivalent to 1/n of the cake, but they also won’t let it get too big because then someone else will call it and all those who didn’t will get a fraction of the cake that is smaller than what they deserve.