Scaled TwoPentomino Rectangles
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.
It has long been known that only four pentominoes can tile rectangles:
For other rectangles that these pentominoes tile, see Mike Reid's
Rectifiable
Polyomino Page.
Here I show which pairs of pentominoes can tile a rectangle,
using the pentominoes at various sizes.
If you find a solution better than one of mine,
please write!
See also Balanced TwoPentomino Rectangles.
I use Solomon W. Golomb's original names for the pentominoes:
This table shows the fewest scaled pentominoes known to be
able to tile a rectangle, using at least one of each pentomino.
The rectangles are not necessarily the smallest that can be tiled,
only the smallest that can be tiled with the fewest tiles.
 F  I  L  N  P  T  U  V  W  X  Y  Z 
F
 *
 11
 4
 ×
 6
 ×
 6
 4
 ×
 ×
 5
 ×

I
 11
 *
 3
 9
 3
 7
 6
 4
 11
 15
 6
 9

L
 4
 3
 *
 4
 3
 8
 4
 4
 4
 5
 6
 7

N
 ×
 9
 4
 *
 4
 10
 4
 4
 ×
 ×
 6
 ×

P
 6
 3
 3
 4
 *
 3
 4
 3
 4
 5
 3
 3

T
 ×
 7
 8
 10
 3
 *
 56
 7
 16
 ×
 4
 ×

U
 6
 6
 4
 4
 4
 56
 *
 3
 ×
 3
 6
 6

V
 4
 4
 4
 4
 3
 7
 3
 *
 30
 ×
 6
 3

W
 ×
 11
 4
 ×
 4
 16
 ×
 30
 *
 ×
 10
 ×

X
 ×
 15
 5
 ×
 5
 ×
 3
 ×
 ×
 *
 5
 ×

Y
 5
 6
 6
 6
 3
 4
 6
 6
 10
 5
 *
 5

Z
 ×
 9
 7
 ×
 3
 ×
 6
 3
 ×
 ×
 5
 *

So far as I know, these solutions
have the fewest possible tiles. They are not necessarily uniquely minimal.
Solutions labeled BH are by Bryce Herdt.
3 Tiles
4 Tiles
5 Tiles
6 Tiles
7 Tiles
8 Tiles
9 Tiles
10 Tiles
11 Tiles
15 Tiles
16 Tiles
30 Tiles
56 Tiles
Jenard Cabilao suggested allowing onecell holes in rectangles.
The holes may not touch the perimeter or one another, not even at corners.
These are holey solutions with fewer tiles than the corresponding
holeless solutions.
2 Tiles
3 Tiles
4 Tiles
5 Tiles
6 Tiles
7 Tiles
8 Tiles
9 Tiles
13 Tiles
Last revised 20220528.
Back to Polyform Tiling
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Polyform Curiosities
Col. George Sicherman
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