- One of the guys, let's say A, cuts the cake into 3 pieces
- B chooses one of the pieces
- C chooses one of the pieces
- A picks the last piece on the table
- A-C, A-B, C-B pairs try to agree that all the pieces are equal. If they cannot agree, they use the "I cut, you choose" procedure for their pieces.
- Party A cuts the cake into 3 pieces regarding just his feelings of equity (step II in fig. 1).
- Party B controls the pieces, if he thinks that at least two of the pieces are good for choosing (means those two are equal), he does nothing.
- Party C picks a piece.
- Party B picks a piece.
- Party A picks the remaining piece.
Everything is perfect. However, if party B thinks that one piece is quite bigger than the other ones, in the second bullet above, he takes the knife and trims the biggest piece in order to produce at least 2 tied pieces. (See the III in fig. 1) Then, no one can pick the "trm" bit in the stage 1. And procedure goes:
- Party C chooses a piece.
- If party C did not choose the trimmed piece (piece number 3 in fig. 1), party B has to pick trimmed piece.
- Party A picks the remaining one.
Stage 1 is done. No one envies the other. Now, it is time to divide the "trm" bit.
Figure 2. Selfridge & Conway stage 2.
- Among B and C, the party who did not pick the trimmed piece in stage 1 cuts the trm bit into 3 by his knife.
- If the party who cuts the trimming into 3 is B, then the parties pick the pieces (t1, t2, t3 in fig. 2) in C, A, B order. If the party that cuts the trm bit is C then the picking order is B, A, C.
Stage 2 ends. It is an envy-free division process. We're all done with this.
The problem, of course, goes too far by considering n parties to share the cake :)) but let us leave the rest of the story to the mathematicians.