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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Polyform Curiosities
Col. George Sicherman
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