The number of pentacube triples, or sets of three different pentacubes,
is *C*(29, 3), or 3654.
Most of these triples can be joined to form symmetric 15-cubes.
These 39 cannot:
AFI,
AIS/AIS′,
AIT,
ASX/AS′X,
ATX,
FGI/FG′I,
GIK/G′IK,
GKX/G′KX,
HIX/H′IX,
HIZ/H′IZ,
IEV/IE′V,
IJW/IJ′W,
IJX/IJ′X,
IKS/IKS′,
IKU,
IQT,
IQZ,
IRX/IR′X,
JXZ/J′XZ,
KLX,
QTX,
QXZ,
RTX/R′TX,
RXZ/R′XZ.

At the other extreme, the triple BLN can form 27086 different symmetric polycubes!

Just 23 triples have unique solutions. These triples are shown below. To see a solution, click on the triple. Cross-sections are shown from top to bottom.

BIK | BIS (BIS′) | FGX (FG′X) | FGZ (FG′Z) |
---|---|---|---|

HIT (H′IT) | HXZ (H′XZ) | IJS (IJ′S′) | IJZ (IJ′Z) |

IRT (IR′T) | IRZ (IR′Z) | KUW | QUX |

RSX (R′S′X) | |||

I am indebted to Gál Péter for his investigation of symmetric pentacube pairs.

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Col. George Sicherman [ HOME | MAIL ]