Cutting cakes and kissing circles

Abstract

To divide a cake into equal sized pieces most people use a knife and a mixture of luck and dexterity. These attempts are often met with varying success. Through precise geometric constructions performed with the knife replacing Euclid's straightedge and without using a compass we find methods for solving certain cake-cutting problems exactly. Since it is impossible to exactly bisect a circular cake when its center is not known, our constructions need to use multiple cakes. Using three circular cakes we present a simple method for bisecting each of them or to find their centers. Moreover, given a cake with marked center we present methods to cut it into n pieces of equal size for n=3,4 and 6. Our methods are based upon constructions by Steiner and Cauer from the 19th and early 20th century.