# Polycube Prisms

An old problem in combinatorial geometry is to find the smallest
solid box that a polycube can tile.
See, for example, Pentacubes in a Box.
Here I show minimal results for a broader problem:
find the smallest prism that a polycube can tile.
Some polycubes that cannot tile boxes can tile prisms.

I do not show flat polycubes, which are prisms in their own right.

The usual definition of a prism is a solid of uniform height over
a polygonal base.
Here the polygon is a polyomino.
I allow polyominoes with holes, though such polyominoes are
technically not polygons.

[ Tetracubes
| Pentacubes
| Hexacubes
]

### Odd Variant

### Holeless Variant

### Odd Variants

These prisms have odd numbers of tiles.
The S pentacube can tile a rectangular prism with dimensions
5×9×15; see Shirakawa's
Box Packing Collection.

If you find an odd variant for another pentacube, please write.

#### Allowing Reflection

### 1 Tile

### 2 Tiles

### 3 Tiles

### 4 Tiles

### 6 Tiles

### 8 Tiles

### Impossible

### Allowing Reflection

Sometimes we can use fewer tiles by letting the hexacube be reflected:

Last revised 2024-02-24.

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Polyform Curiosities

Col. George Sicherman
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