## 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.

### Keywords

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### References

- 1.Bouwkamp, C. J.
*Catalogue of solutions of the rectangular 3×4×5 solid pentomino problem*. 1967. The Netherlands: Technische Hogeschool Eindhoven, Department of Mathematics, Eindhoven.Google Scholar - 2.Bouwkamp, C. J. Packing a rectangular box with the twelve solid pentominoes. 1969.
*J. Combinatorial Theory*7: 278–280.CrossRefGoogle Scholar - 3.Bouwkamp, C. J. Catalogue of solutions of the rectangular 2×5×6 solid pentomino problem. 1978.
*Kon. Med. Akad. Wetensch., Proc, ser A*81: 177–186.Google Scholar