A polycube is a three-dimensional generalization of a polyomino, formed by joining congruent cubes along entire faces, with nonoverlapping interiors. In the counting convention used here, translations and rotations are identified, while distinct mirror images are counted separately. The numbers of polycubes composed of cubes, for , 2, ..., are 1, 1, 2, 8, 29, 166, 1023, ... (OEIS A000162; Ball and Coxeter 1987).

Polycubes may be conveniently represented and visualized in the Wolfram Language using ArrayMesh.

There are 1390 distinct ways to pack the eight tetracubes (polycubes of order ) into a box, identifying whole-box rotations but counting mirror images separately (Beeler 1972). Identifying mirror images as well gives 695 packings.

Demaine and Langerman (2026) reported that deciding whether a single face-connected polycube tiles three-dimensional Euclidean space by translations alone is undecidable. More precisely, the complement of this decision problem is complete among recursively enumerable sets under computable many-one reductions. The authors supply a Lean formalization and disclose language-model assistance with the construction and technical article. The formalization has not been independently rebuilt here.