Fair Shares

The birthday party is over, and one chunk of thickly frosted, richly decorated cake is uneaten. Your mother insists that you and your sister slice the cake into two equal pieces, so that she doesn't have to listen to you fight over which is bigger. What should you do?

The simplest strategy is to let one person cut the cake into two pieces, then let the other person choose first. That's sure to give a result that appears fair to both people.

This divide-and-choose method of settling arguments over the division of cake, goods, or land goes back thousands of years, to biblical times. Mathematicians became interested in the problem of fair division about 60 years ago when they began to wonder what to do when three people want to divide a cake fairly. Getting a fair result for three people turns out to be surprisingly complicated. Here's one way you might do it. Mathematicians call it the "moving knife" strategy.

Suppose that a knife floats above a rectangular cake. Starting at the cake's left end, it moves slowly toward the right. Three people are all told to shout "Stop!" as soon as the knife reaches a point that, in their opinion, is one third of the cake. The first one to shout gets the first piece, and the remaining two people divide the rest using the "I cut, you choose" scheme.

The moving-knife strategy will also work for more than three people. As the knife moves to the right, more and more people drop out with what they think is a fair share until only two people are left to divide the last morsel.

Mathematicians have figured out other methods of sharing things, many of which don't involve a moving knife. Such methods are now used to help handle disputes between people and have even been used among nations to determine offshore mining rights. It's all in the math of fair cake cutting!

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Muse, May/June 1999, p. 28.