Problem: We have a square cake (that is, its horizontal cross-sections are congruent squares). It is frosted evenly on the four sides and the top. How can we cut the cake into n pieces with vertical cuts so that all the pieces have equal amounts of cake and equal amounts of frosting.
Comment: If the cake is cut from the center to an edge, then the problem is reduced to cutting the square top of the cake into n equal areas. For example, when n = 3, the cuts might look like the figure on the right.
Can you solve the problem for n = 2? . . . n = 3? . . . n = 4? other n?
How would you solve the problem if the cake were circular rather than square? Does a similar method work for the square cake?
Look carefully. Does your method for a circular cake have another interpretation in terms of distance along the perimeter of the cake? HINT: Central angle and arc length have the same measure.
Generalize to shapes other than circles and squares.
Extend to n pieces of cake.
Reference: Coxeter, H. M. S. (1969) Introduction to geometry (2nd ed.). New York: Wiley, page 37.