Perfect divisions of a cake

The problem studied in this paper arises from a simple practical problem: how to divide a cake among the children attending a birthday p a r t y i n s u c h a way that each c hild gets the same amount o f c a k e and (perhaps more important to them) the same amount of icing. Let S be a convex set contained in the (xx y)-plane. In mathematical terms, a c a k e C with base S is a solid containing all the points with coordinates (xx y z) such that (xx y 0) 2 S and 0 z h, h > 00 h is called the height o f C. The exposed area of C consists of the boundary of C minus S, i.e. the base of a cake is not considered to be exposed. A cake will be called a polygonal cake if S is a convex polygon. A division of a cake C into k parts by a series of vertical cuts is said to be perfect if each p a r t i s c o n vex and has the same volume and the same exposed area of S. Our birthday c a k e problem can be stated as follows: given a cake C , d o e s i t have a perfect partitioning into k pieces? If a cake has such a partitioning, we will also say that C can be cut perfectly. A c a k e whose base is a square can be cut perfectly into three pieces as follows: take a n y three points x, y and z that divide the perimeter of its base into three pieces of the same length. Now make v ertical cuts along the line segments connecting these points to the center of the base of the cakee see Figure 1. Perfect partitionings of cakes in which t h e v ertical cuts are all along line segments concurrent a t a p o i n t p are called radial perfect partitionings. Notice that for any k > 0, any circular cake C has a radial perfect partitioning into k pieces. This motivates the following deenition. A c a k e C is called graceful if, for every k, there is a perfect radial partitioning of C into k pieces. A natural question arises here: …