Holey 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,
and allowing onecell holes that do not touch the perimeter
or one another, not even at a point.
If you find a solution smaller than one of mine,
please write!
See also
L Shapes from Two Pentominoes.
and
L Shapes from Three Pentominoes.
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 a holey Lshaped polyomino:
 F  I  L  N  P  T  U  V  W  X  Y  Z 
F
 *
 33
 16
 ×
 15
 ×
 10
 21
 ×
 ×
 21
 ×

I
 33
 *
 10
 32
 10
 26
 11
 10
 37
 44
 22
 38

L
 16
 10
 *
 10
 10
 16
 11
 10
 15
 26
 16
 22

N
 ×
 32
 10
 *
 10
 36
 26
 21
 ×
 ×
 26
 ×

P
 15
 10
 10
 10
 *
 10
 10
 10
 20
 26
 10
 10

T
 ×
 26
 16
 36
 10
 *
 23
 27
 62
 ×
 21
 ×

U
 10
 11
 11
 26
 10
 23
 *
 11
 34
 21
 10
 38

V
 21
 10
 10
 21
 10
 27
 11
 *
 11
 ×
 11
 10

W
 ×
 37
 15
 ×
 20
 62
 34
 11
 *
 ×
 21
 ×

X
 ×
 44
 26
 ×
 26
 ×
 21
 ×
 ×
 *
 44
 ×

Y
 21
 22
 16
 26
 10
 21
 10
 11
 21
 44
 *
 25

Z
 ×
 38
 22
 ×
 10
 ×
 38
 10
 ×
 ×
 25
 *

So far as I know, these solutions
have the fewest possible cells.
They are not necessarily uniquely minimal.
10 Cells
11 Cells
15 Cells
16 Cells
20 Cells
21 Cells
22 Cells
23 Cells
25 Cells
26 Cells
27 Cells
32 Cells
33 Cells
34 Cells
36 Cells
37 Cells
38 Cells
44 Cells
62 Cells
Last revised 20230310.
Back to Polyomino and Polyking Tiling
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Polyform Tiling
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Polyform Curiosities
Col. George Sicherman
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