Fair Split

fair split pie in three

It is well known how to split fairly a cake between two people - one of them cuts, the other one picks. The question is, how can you split fairly a cake between three people?

Easy: "Fairly" means that every person gets at least 1/3 of the cake.

Hard: "Fairly" means that every person has the opportunity to get at least as much cake as any other.

Easy (Banach-Knaster method):

The first person cuts 1/3 piece of the cake. If the second person thinks it is larger than 1/3, he can trim it to 1/3. If the third person thinks the cut (and possibly trimmed) piece is larger than 1/3, he can trim it to 1/3 and keep it. Otherwise, the second person takes the piece if he decided to trim it, or the first one, in case he did not. After that, there are two people left, and they can easily split the remaining cake between them. This approach works for any number of people.

Hard (Selfridge-Conway method):

The first person cuts the cake in 3 pieces. The second one takes the biggest piece and trims it so that it becomes as large as the second biggest piece, puts the trimmings aside. The third person picks one of the three big pieces. Then, if the trimmed piece is still available, the second person takes it, if not - he picks whichever he likes. The first person takes the last remaining big piece.

Among the first two people, whoever did not pick the trimmed big piece, splits the trimmings in 3 parts. The other one picks one of these parts, then the first person picks another. The last part goes to the person who split the trimmings.