A polycube is a three-dimensional generalization of a polyomino, formed by joining congruentcubes
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).
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.
composed of 
cubes, for 
, 2, ..., are 1, 1, 2, 8, 29, 166, 1023, ... (OEIS A000162;
Ball and Coxeter 1987).
) into a 
box, identifying whole-box rotations
but counting mirror images separately (Beeler 1972).
Identifying mirror images as well gives 695 packings.
