Suppose you have N people and one cake. How can you cut the cake such that each person is satisfied that the pieces have been distributed fairly? This is an old problem that Martin Gardner wrote up in his column for Scientific American and in the case of two people it is quite simple: One person gets to cut the cake into two and the other person gets to select the piece they want. (But see later for a problem with this.)
But what if there are more than two people? Below the fold, I give Gardner’s explanation on how to do it, starting with the case where N=3, quoted by Walter Stromquist in an issue of The American Mathematical Monthly.
One person moves a large knife slowly over a cake. The cake may be any shape, but knife must move so that the amount of cake on one side continuously increases from zero to the maximum amount. As soon as anyone of the three believes that the knife is in a position to cut a first slice equal to 1/3 of the cake, he/she shouts ‘Cut!’ The cut is made at that instant, and the person who shouted gets the piece. Since he/she is satisfied that he/she got 1/3, he/she drops out of the cutting ritual. In case two or all three shout ‘Cut!’ simultaneously the piece is given to any one of them.
“The remaining two persons are, of course, satisfied that at least 2/3 of the cake remain. The problem is thus reduced to the previous case …
“This clearly generalizes to N persons.
Note that fairness in this case does not mean that each person gets exactly the same amount. It is very likely that the resulting N pieces will not be exactly equal. It is a slightly different concept of fairness that is based on the fact that each person gets to choose the piece they want and if it turns out that the pieces are not exactly equal and they got less than their 1/N share, they have no one to blame but themselves for their poor judgment. They cannot blame bad luck or collusion by the others. If one uses other methods like a lottery, people may still feel aggrieved that they were the victims of circumstance. After all, when it comes to life in general, telling poor people that it was just their bad luck that they happened to be born into poor families and thus they should be satisfied with their lot will hardly convince them that life is fair, nor should it.
But back to this problem, I am not sure if Gardner ever actually tried his solution out on children. But I did and immediately ran into a problem. When my two daughters were little, I encountered this situation of having to divide something between the two of them and, being the nerdy father that I am and always on the look out, often to their annoyance, of sneaking in ways to teach them science, math, logic, and fairness using everyday events, I told them about the ‘one person cuts, the other person chooses’ method.
They both thought about this for a short while and spotted the flaw in the system. The method’s success depends on the ability of the cutter to cut exactly (or very close to) two equal pieces. They each felt that neither of them would be able to cut the cake into two equal pieces, however hard they tried, but that they could easily decide which of the two unequal pieces was the larger. Hence choosing became the clearly better option. Neither one wanted to be the cutter and wanted the other to do it.
I was stymied. There was no way I could think of to resolve this impasse. Of course, one could toss a coin or something to decide who gets to cut but that seemed to defeat the purpose since it brought chance back into it. But I could see no other way and in the end it was decided that I would cut (since I was more likely to be able to get close to two equal pieces) and then toss a coin as to who got to choose first.
Too bad Gardner was not a member of my household. He may have figured out something better.