Pentacube Pair Boxes


A pentacube is a solid made of 5 equal cubes joined face to face. There are 29 pentacubes, distinguishing mirror images.

I indicate mirror images with a prime mark (). For example, S′ denotes the mirror image of pentacube S.

All but two of the pentacubes, G and X, can tile a rectangular prism, or box. All but one pair of pentacubes, G-G′, can jointly tile a box.

Here I show the smallest box known to be tilable with each pair of pentacubes, using at least one copy of each pentacube in the pair. If you find a solution smaller than one shown, or solve an unsolved case, please write.

See also Pentacube Pair Odd Boxes.

Tile Counts

Click on an entry in the table to see the corresponding tiling.

A 6612248126686464686612124046
B  61214684881086610815681210612
E   46616446666468448688126981886
F    1889884128661281664124059
G      ×88241241284886654402481812818200832
H        10688646844686888861486
I       101231283612129669969
J           6841046446686668686
K         84464412166126846
L          8434889444588
M           12663088616121068
N            448684481668
P             4663434533
Q              646468646
R                   28688661210612
S                     181261261586
T                 6121263418
U                  816369
V                   1224123
W                    36812
X                     563
Y                      5

Impossible Case

G and G′ cannot tile even one edge of a box, much less a whole box.

Last revised 2024-02-12.

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Col. George Sicherman [ HOME | MAIL ]