Polynomials and packings: A new proof of de Bruijn's theorem

Abstract

In 1969 de Bruijn published a proof of the following fact: An a × ab × abc brick can be used to pack an A × B × C box if, and only if, the integers A, B, C are in some order a multiple of a, a multiple of ab, and a multiple of abc. We give a quick proof of this result based on the following elementary lemma. The polynomial (x^a - 1)(x^ab - 1)(x^abc - 1) divides (x^A - 1)(x^B - 1)(x^C - 1) if, and only if, the integers A, B, C are in some order a multiple of a, a multiple of ab, and a multiple of abc.