Packing a Box with Bricks
Charles H. Jepsen · Mathematics Magazine · 1991
(See [2]. The problem is C/i on p. 14; a combinatorial solution is given on pp. 99-100.) Suppose the hole in the wall is replaced by a box. If the box is assumed to have a fixed orientation (as in Problem 1 below), packing the box is identical to filling the hole. However, in box-packing it seems natural to take rigid motions into account. That is, we consider two packings to be the same if a rigid motion of the box transforms one packing into the other. (See Problem 2 below.) In this note, we present a solution of Problem 2 based on Burnside's Theorem and our matrix algebra solution of Problem 1. Our approach illustrates an interaction among several areas of undergraduate mathematics: geometry, combinatorics, linear algebra, and group theory.