Selfridge–Conway procedure
The Selfridge–Conway procedure is a discrete procedure that produces an envy-free cake-cutting for three partners. It is named after John Selfridge and John Horton Conway. Selfridge discovered it in 1960, and told it to Richard Guy, who told it to many people, but Selfridge did not publish it. John Conway later discovered it independently, and also never published it. This procedure was the first envy-free discrete procedure devised for three partners, and it paved the way for more advanced procedures for "n" partners (see envy-free cake-cutting). A procedure is envy-free  if each recipient believes that (according to their own measure) no other recipient has received a larger share. The maximal number of cuts in the procedure is five. The pieces are not always contiguous. The Procedure. Suppose we have three players P1, P2 and P3. Where the procedure gives a criterion for a decision it means that criterion gives an optimum choice for the player. It remains to divide the trimmings A2. The trimmed piece A1 has been chosen by either P2 or P3; let's call the player who chose it PA and the other player PB. Analysis. Let 's see why the procedure is envy-free. It must be shown that each player believes that no other player received a larger share. Without loss of generality, we can write (see illustration above): In the following analysis "largest" means "largest according to that player": Generalizations. Note that if all we want is an envy-free division for "a part" of the cake (i.e. we allow free disposal), then we only need to use the first part of the Selfridge–Conway procedure, i.e.: This guarantees that there is no envy. This procedure can be generalized to 4 partners in the  following way: This guarantees that there is no envy. By induction, the procedure can be generalized to "n" partners, the first one dividing the cake to formula_1 equal pieces and the other partners follow by trimming. The resulting division is envy-free. We can apply the same procedure again on the remainders. By doing so an infinite number of times, we get an envy-free division of the "entire" cake. A refinement of this infinite procedure yields a finite envy-free division procedure: the Brams–Taylor procedure.