Most mathematics students, and surely all mathematicians, are familiar with calculus
optimization problems, in particular, with the problem type in which an open-topped
box is to be made by cutting corners from an a b sheet of cardboard and folding
up the sides, and students are asked to maximize the attainable volume (see Figure 1).
Let us call this the box problem. Under what conditions is the volume an integer? Or
the dimensions of the wasted cardboard corners? How about the dimensions of the
cardboard?