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
so that all the pieces have equal amounts of cake and equal amounts
of frosting?
Hints/Solution:
Can you solve the problem for n = 2? . . . n = 3? . . . n = 4?
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?
Comments:
Extensions/Variations:
Generalize to shapes other than circles and squares.
References: Coxeter, Introduction to geometry (2nd ed.),
page 37.