Packing Boxes with Bricks

Brick-packing problems In a packing problem we must arrange a given collection of geometric objects in a nonoverlapping configuration to fill some larger object com pletely. Packing problems can be challenging even when only a few simple shapes are involved. For example, several innocuous-looking, but fiendish 3-dimensional packing problems were devised by J. H. Con way [23]. Tiling problems (2-dimensional packing problems) studied in the ancient world include tangrams in China and a conundrum of Archimedes, whose resolution merited a front page article in the New York Times in 2003 [25]. Jigsaw puzzles are familiar instances of more recent tiling problems. The packing problems most studied by mathematicians concern polyominoes?finite sets of rookwise-connected unit cells in an infinite chessboard?and their generalizations to higher dimensions. In this article, we examine packings of rectangular boxes with rectangular bricks. Even in this basic case the problems that arise are interesting and difficult. We treat ?/-dimensional bricks and boxes, including those whose edge lengths are not integers, and answer the following questions as we introduce our packing theorems and con structions.