First, lets define some terms. A person will consider a piece of cake "fair" if in their opinion it constitutes at least a third of the cake. It is "unfair" if they think it鈥檚 less than a third.
Steinhaus鈥檚 solution goes like this:
We have three people, Alice, Bertha and Claire who wish to divide the cake so that each feels that they have at least a third.
First, Alice cuts the cake into three pieces which she considers are fair.
Next, Bertha and Claire look at the three pieces of cake and decide independently of one another how many they think are fair.
If both think at least two pieces are fair, they each take one, and Alice takes the remainder and everyone is happy.
Similarly, if one of Bertha and Claire thinks that two pieces are fair, but the other thinks that only one is fair, then the person who thinks that only one is fair takes that one; and the other person takes one of the pieces that they considered fair - at least one must remain. Again, Alice takes the third piece, and everyone is happy.
Finally, if both Bertha and Claire consider that only one piece is fair, this means that they both consider two pieces to be unfair. In other words, there must be at least one piece that they both consider to be unfair, or less than a third of the cake. They feed this piece to the hapless A, leaving two pieces of cake which they both think make up more than 2/3 of the original. They then push these two pieces back together, and divide them on the "I cut you choose" principle.
In this way, Steinhaus鈥檚 solution ensures that everyone thinks that they have had at least a third of the cake, so in that sense it is "fair".
However, it鈥檚 not envy-free. For example, if Bertha and Claire end up dividing the two remaining pieces between themselves in a way differently to how Alice originally divided them, Alice will certainly think that one of them has had more cake than her.
Yet another theory >>
