Scaled Pentomino Pairs Tiling a Rectangle with the Four Corner Cells
Removed
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
at various scales
to form a rectangle with the four corner cells removed.
Carl Schwenke and Johann Schwenke improved on two of my solutions.
See also
I use Solomon W. Golomb's original names for the pentominoes:
This table shows the smallest total number of copies
of two scaled pentominoes known to be
able to tile a rectangle with four of its corner cells removed,
using at least one copy of each pentomino.
| F | I | L | N | P | T | U | V | W | X | Y | Z |
|---|
| F
| *
| 12
| 10
| 4
| 4
| 34
| 4
| 10
| 4
| ×
| 10
| ×
|
|---|
| I
| 12
| *
| 6
| 9
| 5
| 12
| 13
| 9
| 19
| 4
| 4
| 12
|
|---|
| L
| 10
| 6
| *
| 6
| 4
| 10
| 4
| 10
| 8
| 13
| 6
| 10
|
|---|
| N
| 4
| 9
| 6
| *
| 4
| 9
| 10
| 16
| 10
| 9
| 4
| 10
|
|---|
| P
| 4
| 5
| 4
| 4
| *
| 4
| 7
| 4
| 4
| 9
| 4
| 7
|
|---|
| T
| 34
| 12
| 10
| 9
| 4
| *
| 10
| 72
| 10
| ×
| 20
| ×
|
|---|
| U
| 4
| 13
| 4
| 10
| 7
| 10
| *
| 40
| 8
| 4
| 8
| 46
|
|---|
| V
| 10
| 9
| 10
| 16
| 4
| 72
| 40
| *
| 12
| ×
| 10
| 13
|
|---|
| W
| 4
| 19
| 8
| 10
| 4
| 10
| 8
| 12
| *
| 9
| 8
| 52
|
|---|
| X
| ×
| 4
| 13
| 9
| 9
| ×
| 4
| ×
| 9
| *
| 4
| ×
|
|---|
| Y
| 10
| 4
| 6
| 4
| 4
| 20
| 8
| 10
| 8
| 4
| *
| 12
|
|---|
| Z
| ×
| 12
| 10
| 10
| 7
| ×
| 46
| 13
| 52
| ×
| 12
| *
|
|---|
So far as I know, these solutions
use as few tiles as possible. They are not necessarily uniquely minimal.
4 Tiles
5 Tiles
6 Tiles
7 Tiles
8 Tiles
9 Tiles
10 Tiles
12 Tiles
13 Tiles
16 Tiles
19 Tiles
20 Tiles
22 Tiles
34 Tiles
40 Tiles
46 Tiles
52 Tiles
72 Tiles
Last revised 2024-02-27.
Back to Polyomino and Polyking Tiling
<
Polyform Tiling
<
Polyform Curiosities
Col. George Sicherman
[ HOME
| MAIL
]