Two-Pentomino Square Frames
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.
The January
2008 issue of Erich Friedman's Math Magic
defined a frame as a square polyomino with a centered square hole.
The problem was to find the frame with least area that could be tiled
with a given polyomino.
Here I study the related problem of finding the smallest frame
that can
be tiled with two pentominoes.
Thanks to Joyce Michel for suggesting this problem.
Bryce Herdt solved a balanced variant.
I use Solomon W. Golomb's original names for the pentominoes:
| | F | I | L | N | P | T | U | V | W | X | Y | Z |
|---|
| F | • | 28 | 8 | — | 12 | — | 16 | 12 | — | — | 8 | — |
|---|
| I | 28 | • | 8 | 12 | 8 | 8 | 24 | 8 | 28 | ? | 8 | 64 |
|---|
| L | 8 | 8 | • | 8 | 8 | 12 | 12 | 8 | 12 | 28 | 8 | 16 |
|---|
| N | — | 12 | 8 | • | 8 | 8 | 16 | 12 | — | — | 8 | — |
|---|
| P | 12 | 8 | 8 | 8 | • | 12 | 8 | 12 | 12 | 12 | 8 | 12 |
|---|
| T | — | 8 | 12 | 8 | 12 | • | 12 | — | 12 | — | 16 | — |
|---|
| U | 16 | 24 | 12 | 16 | 8 | 12 | • | 88 | — | 12 | 12 | — |
|---|
| V | 12 | 8 | 8 | 12 | 12 | — | 88 | • | 64 | — | 8 | 8 |
|---|
| W | — | 28 | 12 | — | 12 | 12 | — | 64 | • | — | 12 | — |
|---|
| X | — | ? | 28 | — | 12 | — | 12 | — | — | • | 24 | — |
|---|
| Y | 8 | 8 | 8 | 8 | 8 | 16 | 12 | 8 | 12 | 24 | • | 16 |
|---|
| Z | — | 64 | 16 | — | 12 | — | — | 8 | — | — | 16 | • |
|---|
So far as I know, these solutions
have minimal area. They are not necessarily uniquely minimal.
8 Tiles
12 Tiles
16 Tiles
24 Tiles
28 Tiles
64 Tiles
88 Tiles
A balanced tiling has equal numbers of the two pentominoes.
Here I do not show tilings from the previous section
if they are already balanced.
12 Tiles
16 Tiles
24 Tiles
32 Tiles
36 Tiles
40 Tiles
64 Tiles
96 Tiles
120 Tiles
544 Tiles
In a contiguous variant, all the pentominoes with a given shape
are connected at edges.
Here I do not show tilings from the first section
if they are already contiguous.
8 Tiles
12 Tiles
16 Tiles
24 Tiles
32 Tiles
40 Tiles
48 Tiles
72 Tiles
Last revised 2023-08-03.
Back to Polyomino and Polyking Tiling
<
Polyform Tiling
<
Polyform Curiosities
Col. George Sicherman
[ HOME
| MAIL
]