Posted by tpc at February 8th, 2005

No one shall expel us from the paradise that Calculus has created.
- tpc

The fact that an analytic solution to number theoretical problems exist is bewildering. And often, the analytic proof is much simpler than the elementary proof. The Prime Number Theorem is the most famous example. Another is the evaluation of \zeta(2) and \zeta(3) in terms of double integrals.

The following is strictly not a number theory problem:
A rectangle is divided into smaller rectangles. Each of the smaller rectangles has the property that at least one of the sides has integer length. Show that the large rectangle has the same property.

I first encountered the problem at kuro5hin via Chapterzero, but apparently it dates back to de Bruijin in the sixties. I couldn’t solve it but as luck would have it, saw a solution in Paul Zeitz’s The Art and Craft of Problem Solving, which provided a reference to an article in American Math Monthly (vol 94) by Stan Wagon which gave 14 proofs, including a one line calculus proof.