# Pentacubes in a Box With All 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, for each pentacube, a minimal known
box from which the cells along all
edges have been removed, tiled by that pentacube.
The cross-sections are shown from top to bottom.
If you find a smaller solution for a pentacube, or solve an unsolved case,
please write.

## Solutions

### A

4 tiles, 4×4×3

### B

4 tiles, 4×4×3

### E

32 tiles, 6×6×6

### F

32 tiles, 6×6×6

### G

96 tiles, 12×12×4

#### With Reflection

32 tiles, 6×6×6

### H

16 tiles, 4×8×4

### I

19 tiles, 7×7×3

### J

12 tiles, 5×5×4

### K

32 tiles, 6×6×6

### L

24 tiles, 6×10×3

### M

4 tiles, 4×4×3

### N

12 tiles, 5×5×4

### P

12 tiles, 5×5×4

### Q

16 tiles, 4×8×4

### R

4 tiles, 4×4×3

### S

96 tiles, 12×12×4

#### With Reflection

64 tiles, 10×10×4

### T

No solution known.

### U

68 tiles, 12×12×3

### V

124 tiles, 12×12×5

### W

64 tiles, 10×10×4

### X

No solution.
### Y

16 tiles, 4×8×4

### Z

No solution known.

Last revised 2022-03-04.

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Col. George Sicherman
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