# Pentacubes in a Box Without Corners

## 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.

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 with the corner cells
removed.
The cross-sections are shown from back to front.
If you find a smaller solution for a pentacube, please write.

## Solutions

### A

2 tiles, 2×3×3

### B

2 tiles, 2×3×3

### E

2 tiles, 2×3×3

### F

20 tiles, 3×6×6

### G

8 tiles, 2×4×6

### H

8 tiles, 3×4×4

### I

9 tiles, 1×7×7

### J

8 tiles, 2×4×6

### K

8 tiles, 2×4×6

### L

4 tiles, 1×4×6

### M

312 tiles, 7×8×28

### N

10 tiles, 1×6×9

### P

4 tiles, 1×4×6

### Q

8 tiles, 2×4×6

### R

8 tiles, 2×4×6

### S

#### With Reflection

8 tiles, 2×4×6

#### Without Reflection

248 tiles, 6×8×26

### T

92 tiles, 3×12×13

### U

2 tiles, 2×3×3

### V

56 tiles, 6×6×8

### W

12 tiles, 1×8×8

### X

1 tile, 1×3×3

### Y

8 tiles, 2×3×8

### Z

88 tiles, 4×8×14

Last revised 2016-02-10.

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Col. George Sicherman
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