# Pentacubes in a Box

## Introduction

A *pentacube* is a solid made of five equal cubes joined
face to face.
There are 23 such figures, not distinguishing reflections and rotations:

The six blue tiles have distinct mirror images.
Kate Jones's systematic names are shown in green.
Donald Knuth's names are shown in red.
In all three nomenclatures, pentacubes that lie all in one plane
are named for the corresponding pentominoes.
(Kate Jones uses Solomon Golomb's names; Donald Knuth uses John Conway's names.)

Can a rectangular prism, or box, be filled with copies of a pentacube?
For each pentacube, here is the smallest known box that it can fill.
Some pentacubes have other solutions with the same volume and different
dimensions.
Torsten
Sillke's polycube box tiling page identifies many boxes besides
the smallest that can be tiled by polycubes.
Toshihiro
Shirakawa's Box Packing Collection
has extensive box tiling data for polycubes, edge-polycubes, and
polyhypercubes.

Most of these constructions are taken from the
text representations in
Michael
Reid's Box Collection.
Most of these solutions are Mike's.
In his collection you can also find solutions for polycubes of other sizes.

Neither the G nor the X pentacube can tile a box.
See below for hybrid solutions.

## Solutions

Cross-sections are shown from back to front.
### A

108 tiles, 4×9×15

*Mike Reid*

This was the first solution found:

144 tiles, 4×9×20

*Helmut Postl*

This pentacube also tiles
a 6×10×10 box (120 tiles)
and an 8×8×10 box (128 tiles).

### B

10 tiles, 2×5×5

### E

4 tiles, 2×2×5

Primes: 2×2×5, 3×4×5, 3×5×5,
3×5×6, 2×5×7, 3×5×7,
2×3×10, 3×3×10, 3×3×15.

Without reflection:
2×2×5,
4×5×5,
5×5×5,
3×5×6,
5×5×7,
3×5×8,
3×5×9,
3×5×10,
3×5×11,
3×5×13.

### F

36 tiles, 3×6×10

*Patrick Hamlyn*

### G

Impossible even with reflection

See below for hybrid solutions.

### H

4 tiles, 2×2×5

### I

1 tile, 1×1×5

Primes: 1×1×5.

### J

6 tiles, 2×3×5

Without reflection: 12 tiles, 3×4×5

### K

18 tiles, 3×5×6

### L

2 tiles, 1×2×5

Primes: 1×2×5, 3×5×5, 1×7×15.

### M

198 tiles, 6×11×15

*Toshihiro Shirakawa*
(白川俊博) (2014)

### N

8 tiles, 2×4×5

### P

2 tiles, 1×2×5

Primes: 1×2×5, 3×3×5, 1×7×15.

### Q

4 tiles, 2×2×5

*Patrick Hamlyn*

### R

28 tiles, 4×5×7

Without reflection: 64 tiles, 5×8×8

*Helmut Postl*

### S

18 tiles, 3×5×6

*Patrick Hamlyn*

Without reflections: 24 tiles, 4×5×6

*Patrick Hamlyn*

### T

60 tiles, 3×10×10

*Frits Göbel*

#### Cubical

*Toshihiro Shirakawa* (白川俊博)

### U

6 tiles, 2×3×5

### V

18 tiles, 3×5×6

### W

36 tiles, 5×6×6

*Torsten Sillke*

### X

Impossible

See below for hybrid solutions.

### Y

10 tiles, 1×5×10

*David Klarner*

### Z

120 tiles, 6×10×10

Wolf

#### Cubical

*Toshihiro Shirakawa* (白川俊博)

Neither the G nor the X pentacube can tile a box.
The two together can.
With reflection: 100 G tiles, 40 X tiles, 7×10×10.

*George Sicherman*

Without reflection: 128 G tiles, 72 X tiles, 10×10×10.

*George Sicherman*

The unreflected G and X pentacubes can also tile a
10×10×12 box, a
10×10×13 box, a
10×10×14 box, and a
10×10×15 box.

Last revised 2020-11-07.

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Col. George Sicherman
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