A *cubic* polycube is a polycube whose cells form
the shape of a cube.

A cube with side *n* has *n*^{3}
cells.
The smallest cubic polycube whose volume is a multiple of 5
is the cube with side 5, shown above.
It has 125 cells, so it can be tiled with 25 pentacubes.

Here I show which pairs of pentacubes can tile the a 5×5×5
cube,
using at least one copy of each pentacube.
A prime mark (**′**) after a letter denotes a mirror image.
For example, **S′** is the mirror image of
**S**.
To see a tiling, click on the corresponding entry in the table below.
Missing entries indicate unsolved cases.
Yellow cells indicate that the tiling is unique.

The
**E**,
**I**,
**J** (and **J′**),
**L**,
**N**,
**P**,
and
**Y** pentacubes can each
tile the 5×5×5 cube alone.
To see such tilings, click on the corresponding index link in the table.

If you solve an unsolved case, please write.

See also Tiling a Cuboctal Polycube with Two Pentacubes, Tiling a Rhonic Polycube with Two Pentacubes, and Pentacube Pair Pyramids.

A | B | E | E′ | F | G | G′ | H | H′ | I | J | J′ | K | L | M | N | P | Q | R | R′ | S | S′ | T | U | V | W | X | Y | Z | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

A | @ | @ | @ | × | – | @ | @ | @ | @ | @ | @ | @ | – | – | @ | @ | @ | @ | @ | × | @ | @ | |||||||

B | @ | @ | × | @ | @ | @ | @ | @ | × | @ | @ | @ | @ | @ | @ | @ | @ | @ | × | @ | @ | ||||||||

E | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | |||

F | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | × | @ | @ | ||||||||||

G | × | × | × | × | @ | @ | @ | @ | × | @ | @ | – | × | × | × | @ | × | @ | @ | × | × | @ | × | ||||||

H | × | @ | @ | @ | @ | @ | @ | @ | @ | – | – | – | @ | @ | @ | @ | @ | @ | × | @ | @ | ||||||||

I | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | |||||||||||||

J | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | |||||||||||

K | @ | @ | @ | @ | @ | @ | – | @ | @ | @ | @ | @ | @ | @ | |||||||||||||||

L | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | ||||||||||||||||

M | @ | @ | @ | × | @ | @ | @ | × | @ | × | @ | × | |||||||||||||||||

N | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | ||||||||||||||||||

P | @ | @ | @ | @ | @ | @ | @ | @ | @ | @ | |||||||||||||||||||

Q | – | @ | @ | @ | @ | @ | @ | @ | @ | ||||||||||||||||||||

R | × | @ | @ | @ | @ | @ | @ | × | @ | @ | |||||||||||||||||||

S | × | @ | @ | @ | @ | × | @ | @ | |||||||||||||||||||||

T | @ | @ | × | × | @ | @ | |||||||||||||||||||||||

U | @ | @ | @ | @ | @ | ||||||||||||||||||||||||

V | @ | × | @ | @ | |||||||||||||||||||||||||

W | × | @ | × | ||||||||||||||||||||||||||

X | @ | × | |||||||||||||||||||||||||||

Y | @ | ||||||||||||||||||||||||||||

Z |

Last revised 2024-03-22.

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Col. George Sicherman [ HOME | MAIL ]