The Austin moving-knife procedures are procedures for equitable division of a cake. They allocate each of n partners, a piece of the cake which this partner values as exactly of the cake. This is in contrast to proportional division procedures, which give each partner at least of the cake, but may give more to some of the partners. When , the division generated by Austin's procedure is an exact division and it is also envy-free. Moreover, it is possible to divide the cake to any number k of pieces which both partners value as exactly 1/k. Hence, it is possible to divide the cake between the partners in any fraction (e.g. give 1/3 to Alice and 2/3 to George). When , the division is neither exact nor envy-free, since each partner only values his own piece as , but may value other pieces differently. The main mathematical tool used by Austin's procedure is the intermediate value theorem (IVT). Two partners and half-cakes The basic procedures involve partners who want to divide a cake such that each of them gets exactly one half. Two knives procedure For the sake of description, call the two players Alice and George, and assume the cake is rectangular. Alice places one knife on the left of the cake and a second parallel to it on the right where she judges it splits the cake in two. Alice moves both knives to the right in a way that the part between the two knives always contains half of the cake's value in her eyes (while the physical distance between the knives may change). George says "stop!" when he thinks that half the cake is between the knives. How can we be sure that George can say "stop" at some point? Because if Alice reaches the end, she must have her left knife positioned where the right knife started. The IVT establishes that George must be satisfied the cake is halved at some point. A coin is tossed to select between two options: either George receives the piece between the knives and Alice receives the two pieces at the flanks, or vice versa. If partners are truthful, then they agree that the piece between the knives has a value of exactly 1/2, and so the division is exact. One knife procedure A single knife can be used to achieve the same effect. Alice rotates the knife over