Tiling a Polyomino at Scale 2 with Two Pentominoes
A pentomino is a figure made of five squares joined
edge to edge.
There are 12 such figures, not distinguishing reflections and rotations.
They were first enumerated and studied by Solomon Golomb.
Here I study the problem of arranging copies of two pentominoes
to form some polyomino that has been scaled up by a factor of 2.
See also
Andrew Bayly contributed tilings.
This table shows the smallest total number of copies
of two pentominoes known to be
able to tile some polyomino enlarged by a scale factor of 2,
using at least one copy of each pentomino.
The blue indexes are links to tilings by the specified pentomino alone.
| F | I | L | N | P | T | U | V | W | X | Y | Z |
|---|
| F | *
| 12
| 8
| 64
| 8
| 16
| 8
| 8
| ?
| ?
| 8
| 8
|
|---|
| I | 12
| *
| 4
| 8
| 4
| 8
| 8
| 8
| 12
| 20
| 8
| 8
|
|---|
| L | 8
| 4
| *
| 4
| 4
| 4
| 8
| 4
| 8
| 12
| 8
| 4
|
|---|
| N | 64
| 8
| 4
| *
| 4
| 12
| 8
| 8
| 64
| 128
| 8
| 8
|
|---|
| P | 8
| 4
| 4
| 4
| *
| 8
| 4
| 4
| 8
| 8
| 4
| 8
|
|---|
| T | 16
| 8
| 4
| 12
| 8
| *
| 16
| 8
| 16
| ?
| 8
| 16
|
|---|
| U | 8
| 8
| 8
| 8
| 4
| 16
| *
| 16
| 16
| 12
| 8
| 8
|
|---|
| V | 8
| 8
| 4
| 8
| 4
| 8
| 16
| *
| 16
| ?
| 8
| 4
|
|---|
| W | ?
| 12
| 8
| 64
| 8
| 16
| 16
| 16
| *
| ?
| 8
| 8
|
|---|
| X | ?
| 20
| 12
| 128
| 8
| ?
| 12
| ?
| ?
| *
| 16
| ?
|
|---|
| Y | 8
| 8
| 8
| 8
| 4
| 8
| 8
| 8
| 8
| 16
| *
| 8
|
|---|
| Z | 8
| 8
| 4
| 8
| 8
| 16
| 8
| 4
| 8
| ?
| 8
| *
|
|---|
So far as I know, these solutions
use as few tiles as possible. They are not necessarily uniquely minimal.
4 Tiles
8 Tiles
12 Tiles
16 Tiles
20 Tiles
64 Tiles
128 Tiles
Last revised 2025-05-13.
Back to Polyomino and Polyking Tiling
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Polyform Curiosities
Col. George Sicherman
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