Packing boxes with harmonic bricks

Given a box of integral dimensions and a supply of bricks all having the same integral dimensions, what is the largest number of bricks that can be packed in the box with the sides of the bricks parallel to the sides of the box? In this general form the question is very difficult. We can think of the box as being made up of unit cells and say that a set R of cells is a representing set for a brick of given dimensions provided every brick that can be placed in the box, sides parallel to the box, contains at least one of the cells in R. The maximum number of bricks that can be placed in the box is then less than or equal to the minimum cardinality of a representing set. In general, there is not equality. In the case of two dimensions and a harmonic brick, we prove there is equality always and exhibit a best packing. For three-dimensional boxes and a harmonic brick, there need not be equality. We derive several results which are of the nature that if certain inequalities relating the dimensions of the box and the brick are satisfied, then equality occurs. Our results are strong enough to imply, for example, that if the smallest face of the brick packs each face of the box perfectly, then there is equality. For a 1 × 2 × 4 brick, there is always equality if one of the dimensions of the box is even.