L Shapes from 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 forming an Lshaped (sixsided) polyomino
with copies of two pentominoes, using at least one of each.
If you find a smaller solution than one of mine,
please write!
See also
L Shapes from Three Pentominoes
and
L Shapes from Pentacube Pairs.
I use Solomon W. Golomb's original names for the pentominoes:
This table shows the smallest total number of two pentominoes known to be
able to tile an Lshaped polyomino:
 F  I  L  N  P  T  U  V  W  X  Y  Z 
F
 *
 17
 5
 ×
 3
 ×
 2
 6
 ×
 ×
 5
 ×

I
 17
 *
 2
 7
 2
 17
 40
 2
 32
 ×
 5
 26

L
 5
 2
 *
 2
 2
 9
 3
 2
 3
 7
 5
 9

N
 ×
 7
 2
 *
 2
 12
 6
 8
 ×
 ×
 8
 ×

P
 3
 2
 2
 2
 *
 2
 2
 2
 4
 6
 2
 2

T
 ×
 17
 9
 12
 2
 *
 46
 ×
 15
 ×
 8
 ×

U
 2
 40
 3
 6
 2
 46
 *
 58
 ×
 6
 2
 ×

V
 6
 2
 2
 8
 2
 ×
 58
 *
 43
 ×
 8
 2

W
 ×
 32
 3
 ×
 4
 15
 ×
 43
 *
 ×
 8
 ×

X
 ×
 ×
 7
 ×
 6
 ×
 6
 ×
 ×
 *
 11
 ×

Y
 5
 5
 5
 8
 2
 8
 2
 8
 8
 11
 *
 5

Z
 ×
 26
 9
 ×
 2
 ×
 ×
 2
 ×
 ×
 5
 *

So far as I know, these solutions
have the fewest possible tiles.
They are not necessarily uniquely minimal.
2 Tiles
3 Tiles
4 Tiles
5 Tiles
6 Tiles
7 Tiles
8 Tiles
9 Tiles
11 Tiles
12 Tiles
15 Tiles
17 Tiles
26 Tiles
32 Tiles
40 Tiles
43 Tiles
46 Tiles
58 Tiles
Last revised 20210320.
Back to Polyomino and Polyking Tiling
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Polyform Curiosities
Col. George Sicherman
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