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# Can you complete this technique for fairly dividing something among 3 people?

I just learned about this and decided to share it in the form of a problem. Most of you probably remember from childhood the technique for splitting something, like a candy bar, between two people. One person cuts the thing into two parts and the other has first choice of which to take. Each person ends up believing that he got at least as much as the other.

About 50 years ago mathematicians worked on the problem of determining how to fairly divide something among 3 people so that each person is convinced he got as much as anybody else. There are ways of extending the technique to more than 3 people, but it gets a bit involved. The problem of dividing something among n people so that each person thinks he got at least 1/n is much simpler, but I will not be describing it here.

I will start describing the technique. See if you can complete it. Designate the 3 people as A, B and C. A divides a cake (or pizza if you prefer) into 3 equal (from his point of view) slices. If B thinks that the two larger pieces are the same size then C chooses next, then B and then A. In this case we are done since everyone should be satisfied he has as much as anyone else.

If B thinks the largest piece is bigger than the next largest piece, he cuts a portion of the largest piece so that he thinks the two largest pieces are the same size. The cutting is put on the side for the next stage. Again the order of choosing is C, B and A. The only constraint is that if C does not choose the cut piece then B must choose it. At this point everyone thinks he has at least as much as anyone else.

For the second stage, the piece on the side will be divided in 3 parts and each person chooses one of the parts. Who should do the cutting and in what order should the parts be chosen so that each person feels he ended up with at least as much as anyone else?

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