Prime Boxes for the K and S Pentacubes

A pentacube is a solid made of 5 equal cubes joined face to face. There are 23 different pentacubes, identifying mirror images, or 29, distinguishing mirror images. These are pentacubes K and S:

A prime box for a set of polycubes is a rectangular prism that the polycubes can tile jointly, that cannot be dissected into smaller rectangular prisms that the polycubes can tile. A polycube can have at most finitely many prime boxes.

Here I show some prime boxes for pentacubes K and S used together. I require the tilings to use at least one copy of each pentacube. Click on a tiling's dimensions to see a picture of the tiling.

If you find any new prime boxes, please write.

TilesDimensionsTilings
122×5×631
183×5×620 854
273×5×9?
322×8×10?
4×4×10?
363×4×15?
402×10×10?
542×9×15?
842×7×30?
1263×7×30?
Impossible
1×M×N
2×2×N
2×3×N
2×4×N
2×5×N,
N≢0 (mod 6)
2×7×10
2×7×15
2×9×10
3×3×N
3×4×5
3×4×10
3×5×5
3×5×7
3×5×8
4×4×5
4×5×5

Last revised 2026-06-16.

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Col. George Sicherman [ HOME | MAIL ]