Pentacube Compatibility

A pentacube is a solid made of five cubes joined face to face. There are 23 pentacubes, not distinguishing reflections and rotations:

The six blue tiles have left- and right-handed forms.

Kate Jones's systematic names are shown in green. The mirror forms of V1, S1–S2, and L1–L4 are called V2, N1–N2, and J1–J4. But L3 and J3 are identical because they have mirror symmetry through a plane diagonal.

Donald Knuth's systematic names are shown in red.

Here are the least known numbers of tiles needed to construct a solid that can be tiled with either of two pentacubes, not letting the pentacubes be reflected. Every pair of pentacubes is compatible, using at most 14 tiles. If you find a compatibility smaller than one shown here, please write.

Click on a number to see the tiling.

See also Pentacube Odd Pairs.

  ABEE′FGG′HH′IJJ′KLMNPQRR′SS′TUVWXYZ
A 2223282224222324242222
B  222242422222242224224
E   22222242222222232222224222
F    2252222222222442222
G      224102222422222244442622
H        242222422222224242822
I       382102241064545525
J           222422222222224422
K         22222222222822
L          4222222222822
M           422242664224
N            22232222822
P             2222222422
Q              224232622
R                   2234442422
S                     23222622
T                 326422
U                  22224
V                   61424
W                    624
X                     210
Y                      2
Z                       

Last revised 2026-06-29.

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Col. George Sicherman [ HOME | MAIL ]