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 [ HOME | MAIL ]