the cake through 180° keeping a half on either side. George says "stop!" when he agrees. Alice must of course end the turn with the knife on the same line as where it started. Again, by the IVT, there must be a point in which George feels that the two halves are equal. Two partners and general fractions As noted by Austin, the two partners can find a single piece of cake that both of them value as exactly , for any integer . Call the above procedure : Alice makes parallel marks on the cake such that pieces so determined have a value of exactly . If there is a piece that George also values as , then we are done. Otherwise, there must be a piece that George values as less than , and an adjacent piece that George values as more than . Let Alice place two knives on the two marks of one of these pieces, and move them in parallel, keeping the value between them at exactly , until they meet the marks of the other piece. By the IVT, there must be a point at which George agrees that the value between the knives is exactly . By recursively applying , the two partners can divide the entire cake to pieces, each of which is worth exactly for both of them: Use to cut a piece which is worth exactly for both partners. Now the remaining cake is worth exactly for both partners; use to cut another piece worth exactly for both partners. Continue like this until there are pieces. Two partners can achieve an exact division with any rational ratio of entitlements by a slightly more complicated procedure. Many partners By combining with the Fink protocol, it is possible to divide a cake to partners, such that each partner receives a piece worth exactly for him: Partners #1 and #2 use to give each one of them a piece worth exactly 1/2 for them. Partner #3 uses with partner #1 to get exactly 1/3 of his share and then with partner #2 to get exactly 1/3 of her share. The first piece is worth exactly 1/6 for partner #1 and so partner #1 remains with exactly 1/3; the same is true for partner #2. As for partner #3, while each piece can be more or less than 1/6, the sum of the two