Packing Handed Pentacubes

Abstract

To refresh the reader’s memory, there are 29 distinct three-dimensional pentacubes; a pentacube being a solid built from five unit cubes such that neighbor cubes have a full face in common and any component cube has at least one neighbor. Of them, the 12 planar pentacubes, also called solid pentominoes, are well known in the field of recreational mathematics. Together they have a volume count of 60 units, and boxes of various sizes such as (2 × 3 × 10), (2×5×6), (3×4×5) can be packed with them [1,2,3]. Among the non-planar pentacubes there are 5 that have at least one plane of symmetry; each of them is its own mirror image. The remaining 12 pentacubes consist of 6 pairs, each pair containing a pentacube and its mirror image. The pentacubes of a pair are in the same relation to each other as one’s left and right hands; I shall, therefore, call them a pair of handed pentacubes. The complete set of handed (or chiral) pentacubes is depicted in Figure 1, together with their identifying labels 1 through 12. Again, their total volume count is 60 units and, as with the planar pentacubes, it is natural to ask whether they can pack a 3 × 4 × 5 box, for example.