Diophantine boxes

Abstract

I have a soft spot for geometric problems which lead to diophantine equations, that is, equations for which only positive integer solutions are sought. I have not seen the following naturally occurring problem discussed anywhere. Problem A A closed rectangular box has its volume numerically equal to its total surface area. If the length, breadth and height of the box are all positive integers, find all possible sets of dimensions of the box. The solution of this pretty problem is completely elementary and can be fully understood by those with a basic knowledge of algebra. We remark that to solve any three-dimensional problem it is often useful to step down a dimension and to consider the corresponding problem for plane figures.