Tiling a Polycube at Scale 2 with a Pentacube

Introduction

A polycube is a solid made of equal cubes joined face to face, and a pentacube is a polycube with 5 cells. There are 29 pentacubes, distinguishing mirror images:

Any polycube may be enlarged by a scale factor of 2 to form a new polycube with 8 times as many cells as the original polycube. Here I show the smallest known such polycubes that can be tiled with a given pentacube. Chiral pentacubes may not be reflected when used in these tilings.

If you find a smaller solution, please write.

See also Tiling a Polyomino at Scale 2 with a Pentomino.

Tilings

For each chiral pair I show a tiling for only one member of the pair.

The G and X pentacubes cannot tile any polycube scaled up by 2.

The solution shown for pentacube W is a rectangular box tiling. Smaller solutions may exist.

8 Tiles

16 Tiles

24 Tiles

32 Tiles

40 Tiles

48 Tiles

72 Tiles

96 Tiles

Last revised 2026-04-21.

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Col. George Sicherman [ HOME | MAIL ]