Initially, each partner makes a vertical cut such that the cake to its left is worth for him exactly . The leftmost cut is selected. Suppose this cut belongs to Alice. So Alice receives the leftmost piece and her value is exactly . The remainder has to be divided between the remaining partners (Bob and Carl). Alice's part is worth
at most and the remainder is worth
at least for Bob and Carl. So, if Bob and Carl each receive at least half of the remainder, they do not envy. The challenge is to make sure Alice won't envy any of them. The solution is based on the following observation:
For each angle \alpha\in[0,180^\circ], Alice can put a knife in angle \alpha and cut the remainder to two halves equal in her eyes. This means that Alice can rotate a knife over the remainder such that the parts from the two sides of the knife are always equal in her eyes. When the knife is at angle 0, Bob (weakly) prefers either the piece above the knife or the piece below the knife; when the knife is at angle 180, the pieces are reversed. Hence, by the
intermediate value theorem, there must be an angle in which Bob thinks the pieces from both sides of the knife are equal. At this angle, Bob shouts "stop!". The cake is cut, Carl chooses a piece and Bob receives the other piece. == Analysis ==