# Pentacubes in a Box With Four Edges Removed

## 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 names are shown in green.
Donald Knuth's names are shown in red.

In Kate Jones's nomenclature and mine, flat pentacubes
bear Solomon Golomb's names for the corresponding pentominoes.
In Donald Knuth's nomenclature, flat pentacubes bear
John Conway's names for the corresponding pentominoes.

All but two pentacubes can tile a rectangular prism, or box;
see Pentacubes in a Box.
Here I show that every pentacube can tile a box from which the cells along four
parallel edges have been removed.
Dimensions are given with the height last.
The cross-sections are shown from top to bottom.
If you find a smaller solution for a pentacube, please write.

## Solutions

### A

2 tiles, 3×3×2

### B

2 tiles, 3×3×2

### E

2 tiles, 3×3×2

### F

20 tiles, 3×8×5

### G

8 tiles, 4×6×2

### H

8 tiles, 4×6×2

### I

5 tiles, 3×3×5

### J

8 tiles, 4×6×2

### K

8 tiles, 4×6×2

### L

4 tiles, 4×6×1

### M

20 tiles, 4×6×5

### N

10 tiles, 6×9×1

### P

4 tiles, 4×6×1

### Q

8 tiles, 4×6×2

### R

8 tiles, 4×6×2

### S

16 tiles, 4×6×4

#### With Reflection

8 tiles, 4×6×2

### T

112 tiles, 12×12×4

### U

2 tiles, 3×3×2

### V

16 tiles, 4×6×4

### W

12 tiles, 8×8×1

### X

1 tile, 3×3×1

### Y

8 tiles, 3×8×2

### Z

140 tiles, 12×12×5

Last revised 2022-03-03.

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Col. George Sicherman
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