I indicate mirror images with a prime mark (**′**).
For example, **S′** denotes the mirror image of
pentacube **S**.

All but two of the pentacubes, **G** and **X**,
can tile a rectangular prism, or *box.*
A box is *odd* if it has an odd number of cells.
Such a box must have all its dimensions odd.

Here I show the smallest odd box known to be tilable with each pair of pentacubes, using at least one copy of each of the two pentacubes. If you find a solution smaller than one shown, or solve an unsolved case, please write.

See also Pentacube Pair Boxes.

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 | 25 | 9 | 15 | ? | 45 | 21 | 15 | 15 | 15 | 25 | 15 | 9 | 45 | ? | 15 | 15 | 15 | 15 | 21 | ? | 15 | 15 | |||||||

B | 15 | 21 | 49 | 45 | 15 | 9 | 15 | 15 | 35 | 15 | 9 | 15 | 25 | 21 | 15 | 9 | 15 | 21 | 49 | 15 | 21 | ||||||||

E | 15 | 15 | 15 | 21 | 15 | 15 | 9 | 9 | 9 | 9 | 9 | 9 | 15 | 9 | 9 | 9 | 15 | 9 | 15 | 15 | 9 | 9 | 9 | 21 | 15 | 15 | |||

F | 21 | 15 | 9 | 15 | 15 | 7 | 21 | 15 | 9 | 15 | 21 | 15 | 21 | 9 | 9 | 21 | 63 | 5 | 9 | ||||||||||

G | × | 45 | ? | 27 | 15 | 9 | 21 | 15 | ? | 21 | 9 | 45 | ? | ? | 33 | 21 | 27 | 15 | 15 | 33 | ? | 15 | 39 | ||||||

H | 45 | 9 | 9 | 9 | 9 | 9 | 21 | 15 | 9 | 45 | 45 | 45 | 15 | 15 | 15 | 15 | 9 | 9 | 35 | 15 | 15 | ||||||||

I | 15 | 21 | 3 | 15 | 15 | 3 | 9 | 15 | 21 | 9 | 9 | 7 | 9 | 9 | 7 | 9 | |||||||||||||

J | 9 | 9 | 9 | 15 | 9 | 9 | 9 | 9 | 15 | 9 | 9 | 9 | 9 | 9 | 9 | 21 | 9 | 9 | |||||||||||

K | 15 | 21 | 15 | 9 | 9 | 15 | 27 | 21 | 9 | 15 | 9 | 15 | 15 | 15 | |||||||||||||||

L | 15 | 7 | 3 | 9 | 9 | 15 | 9 | 9 | 9 | 9 | 5 | 9 | 9 | ||||||||||||||||

M | 21 | 9 | 15 | ? | 15 | 21 | 9 | 21 | 21 | × | 15 | 21 | |||||||||||||||||

N | 9 | 15 | 21 | 15 | 15 | 9 | 15 | 21 | 25 | 21 | 15 | ||||||||||||||||||

P | 9 | 9 | 9 | 3 | 9 | 3 | 9 | 5 | 3 | 3 | |||||||||||||||||||

Q | 45 | 9 | 9 | 15 | 9 | 9 | 15 | 15 | 9 | ||||||||||||||||||||

R | 45 | 21 | 21 | 15 | 9 | 15 | 21 | 117 | 15 | 15 | |||||||||||||||||||

S | 27 | 21 | 15 | 21 | 21 | 15 | 15 | 15 | |||||||||||||||||||||

T | 9 | 15 | 21 | 55 | 9 | 25 | |||||||||||||||||||||||

U | 9 | 21 | 3 | 11 | 9 | ||||||||||||||||||||||||

V | 21 | 49 | 15 | 3 | |||||||||||||||||||||||||

W | 57 | 15 | 15 | ||||||||||||||||||||||||||

X | 5 | 63 | |||||||||||||||||||||||||||

Y | 5 | ||||||||||||||||||||||||||||

Z |

**M** and **X**, colored like a 3D checkerboard,
have four cells of one color and one of the other.
Thus an odd number of **M** and **X** pentacubes
have a net color imbalance that is an odd multiple of 3.
Every odd box has a color imbalance of 1.

Last revised 2024-02-05.

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Col. George Sicherman [ HOME | MAIL ]