POLYOMINO TILINGS

Polyomino Tilings

Select polyominoes for a set (currently 1 or 2), for which tilings should be shown.

Then click "Show" button.

You may also see list of all polyomino sets for which data is available here.


Pentominoes

Prime rectangles: ≥ 27.

Smallest rectangle tilings

Smallest rectangles (3x20, 4x15, 5x12, 6x10):

Smallest square (10x10):

Rectangle tilings' solutions count (including symmetric)

Blue number - strongly prime rectangle (which cannot be divided into two or more number of rectangles tileable by this set).

Green number - weakly prime rectangle (which cannot be divided into two rectangles tileable by this set, but which can be divided into three or more rectangles).

Purple number - prime rectangle (unknown if weakly or strongly prime).

Red number - composite rectangle (which can be divided into two rectangles tileable by this set).

Gray number - it is unknown whether rectangle is prime or composite.

Question mark (?) - solution count is unknown.

Click on underlined numbers to view picture with one solution.

w \ h
1-2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
N>0
1-2
0
3
0
0
4
0
0
0
5
0
0
0
0
6
0
0
0
0
0
7
0
0
0
0
0
0
8
0
0
0
0
0
0
0
9
0
0
0
0
0
0
0
0
10
0
0
0
0
9356
≥1
≥1
≥1
≥1
11
0
0
0
0
0
0
0
0
≥1
0
12
0
0
0
4040
0
0
0
0
≥1
0
0
13
0
0
0
271280
0
0
0
0
≥1
0
0
0
14
0
0
0
8298320
0
0
0
0
≥1
0
0
0
0
15
0
0
1472
183598324
2.05208039×10¹³
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
16
0
0
0
3.33641767×10¹⁰
0
0
0
0
≥1
0
0
0
0
≥1
0
17
0
0
0
5.25204609×10¹¹
0
0
0
0
≥1
0
0
0
0
≥1
0
0
18
0
0
0
7.44245624×10¹²
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
19
0
0
0
9.71582726×10¹³
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
20
0
8
540749904
1.18879305×10¹⁵
6.65070500×10¹⁹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
21
0
0
0
1.37948944×10¹⁶
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
≥1
?
22
0
0
0
1.53193736×10¹⁷
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
≥1
?
23
0
0
0
1.63956824×10¹⁸
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
≥1
?
24
0
0
0
1.70043388×10¹⁹
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
≥1
?
25
0
229344
9.39808083×10¹³
1.71652106×10²⁰
≥1.84467440×10²⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
26
0
0
0
≥1.84467440×10²⁰
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
≥1
?
27
0
0
0
≥1.84467440×10²⁰
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
≥1
?
28
0
0
0
≥1.84467440×10²⁰
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
≥1
?
29
0
0
0
≥1.84467440×10²⁰
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
≥1
?
30
0
385887080
6.81049507×10¹⁷
≥1.84467440×10²⁰
≥1.84467440×10²⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
31
0
0
0
≥1.84467440×10²⁰
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
≥1
?
32
0
0
0
≥1.84467440×10²⁰
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
≥1
?
33
0
0
0
≥1.84467440×10²⁰
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
≥1
?
34
0
0
0
≥1.84467440×10²⁰
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
≥1
?
35
0
2.60555919×10¹²
≥1.84467440×10²⁰
≥1.84467440×10²⁰
≥1.84467440×10²⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
36
0
0
0
≥1.84467440×10²⁰
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
≥1
?
37
0
0
0
≥1.84467440×10²⁰
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
≥1
?
38
0
0
0
≥1.84467440×10²⁰
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
≥1
?
39
0
0
0
≥1.84467440×10²⁰
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
≥1
?
40
0
1.10821096×10¹⁵
≥1.84467440×10²⁰
≥1.84467440×10²⁰
≥1.84467440×10²⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
N>0
x
5k
5k
all
5k
5k
5k
5k
all
5k
?
?
?
?
?
?
?
?
?

Smallest prime reptiles

Smallest prime reptiles (5Ix4, 5Lx4, 5Nx4, 5Px4, 5Rx4, 5Tx4, 5Ux4, 5Vx4, 5Wx4, 5Xx4, 5Yx4, 5Zx4):

Reptile tilings' solutions count (including symmetric)

polyomino \ n²
1²
2²
3²
4²
5²
6²
7²
8²
9²
I pentomino
0
0
0
540749904
17165210687688711858
≥18446744073709551615
≥1
≥1
≥1
L pentomino
0
0
0
≥1
≥1
≥1
≥1
≥1
≥1
N pentomino
0
0
0
≥1
≥1
≥1
≥1
≥1
≥1
P pentomino
0
0
0
≥1
≥1
≥1
≥1
≥1
≥1
R pentomino
0
0
0
≥1
≥1
≥1
≥1
≥1
≥1
T pentomino
0
0
0
≥1
≥1
≥1
≥1
≥1
≥1
U pentomino
0
0
0
≥1
≥1
≥1
≥1
≥1
≥1
V pentomino
0
0
0
≥1
≥1
≥1
≥1
≥1
≥1
W pentomino
0
0
0
≥1
≥1
≥1
≥1
≥1
≥1
X pentomino
0
0
0
≥1
≥1
≥1
≥1
≥1
≥1
Y pentomino
0
0
0
≥1
≥1
≥1
≥1
≥1
≥1
Z pentomino
0
0
0
≥1
≥1
≥1
≥1
≥1
≥1

See Also

Z tetrominoI pentomino