Here I show minimal results for a broader problem: find the smallest prism that a polycube can tile. Some polycubes that cannot tile boxes can tile prisms.
I do not show flat polycubes, which are prisms in their own right.
The usual definition of a prism is a solid of uniform height over a polygonal base. Here the polygon is a polyomino. I allow polyominoes with holes, though such polyominoes are technically not polygons.
[ Tetracubes | Pentacubes | Hexacubes ]
The S pentacube can tile a rectangular prism with dimensions 5×9×15; see Shirakawa's Box Packing Collection.
If you find an odd variant for another pentacube, please write.
Last revised 2022-09-07.