# 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.
In both nomenclatures polycubes that lie all in one plane
are named for the corresponding polyominoes.

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

### C

8 tiles, 2×4×6

### E

2 tiles, 2×3×3

### F

20 tiles, 3×6×6

### 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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