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POLYOMINO TILINGS

Polyomino Tilings

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

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


F hexomino

Area: 6.

Size: 2x4.

Holes: 0.

Order: 2.

Square order: 24.

Prime rectangles: ≥ 1.

Smallest rectangle tilings

Smallest rectangle (3x4):

Smallest square (12x12):

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
2
0
5
0
0
0
0
6
0
0
4
0
0
7
0
0
0
0
0
0
8
0
4
0
0
16
0
0
9
0
0
8
0
0
0
64
0
10
0
0
0
0
0
0
0
0
0
11
0
0
0
0
0
0
0
0
0
0
12
0
8
16
0
64
288
256
512
3584
7296
8192
13
0
0
0
0
0
0
0
0
0
0
38912
0
14
0
0
0
0
0
0
0
0
0
0
125952
0
?
15
0
0
32
0
0
0
1024
0
0
0
193024
0
?
?
16
0
16
0
0
256
0
0
4096
0
0
458752
0
?
≥1
?
17
0
0
0
0
0
0
0
0
0
0
1753088
0
?
?
?
?
18
0
0
64
0
0
0
4096
0
0
0
3833856
0
?
?
≥1
?
?
19
0
0
0
0
0
0
0
0
0
0
7063552
0
?
?
?
?
?
?
20
0
32
0
0
1024
0
0
32768
0
0
22675456
0
?
≥1
?
?
≥1
?
?
21
0
0
128
0
0
0
16384
0
0
0
63242240
0
?
?
≥1
?
?
?
≥1
?
22
0
0
0
0
0
0
0
0
0
0
125124608
0
?
?
?
?
?
?
?
?
23
0
0
0
0
0
0
0
0
0
0
310321152
0
?
?
?
?
?
?
?
?
24
0
64
256
0
4096
82944
65536
262144
15253504
53231616
930086912
2.05498777×10¹⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
25
0
0
0
0
0
0
0
0
0
0
2.15364403×10¹⁰
0
?
?
?
?
?
?
?
?
26
0
0
0
0
0
0
0
0
0
0
4.73969459×10¹⁰
0
?
?
?
?
?
?
?
?
27
0
0
512
0
0
0
262144
0
0
0
1.31950510×10¹¹
0
?
?
≥1
?
?
?
≥1
?
28
0
128
0
0
16384
0
0
2097152
0
0
3.43796613×10¹¹
0
?
≥1
?
?
≥1
?
?
?
29
0
0
0
0
0
0
0
0
0
0
7.69429340×10¹¹
0
?
?
?
?
?
?
?
?
30
0
0
1024
0
0
0
1048576
0
0
0
1.92346324×10¹²
0
?
?
≥1
?
?
?
≥1
?
31
0
0
0
0
0
0
0
0
0
0
5.19077756×10¹²
0
?
?
?
?
?
?
?
?
32
0
256
0
0
65536
0
0
16777216
0
0
1.24361113×10¹³
0
?
≥1
?
?
≥1
?
?
?
33
0
0
2048
0
0
0
4194304
0
0
0
2.93979606×10¹³
0
?
?
≥1
?
?
?
≥1
?
34
0
0
0
0
0
0
0
0
0
0
7.69620810×10¹³
0
?
?
?
?
?
?
?
?
35
0
0
0
0
0
0
0
0
0
0
1.94859609×10¹⁴
0
?
?
?
?
?
?
?
?
36
0
512
4096
0
262144
23887872
16777216
134217728
6.51416043×10¹¹
3.88377870×10¹²
4.61902928×10¹⁴
1.07401547×10¹⁵
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
37
0
0
0
0
0
0
0
0
0
0
1.15441238×10¹⁵
0
?
?
?
?
?
?
?
?
38
0
0
0
0
0
0
0
0
0
0
2.97501684×10¹⁵
0
?
?
?
?
?
?
?
?
39
0
0
8192
0
0
0
67108864
0
0
0
7.26020585×10¹⁵
0
?
?
≥1
?
?
?
≥1
?
40
0
1024
0
0
1048576
0
0
1.07374182×10¹⁰
0
0
1.76774465×10¹⁶
0
?
≥1
?
?
≥1
?
?
?
41
0
0
0
0
0
0
0
0
0
0
4.50069129×10¹⁶
0
?
?
?
?
?
?
?
?
42
0
0
16384
0
0
0
268435456
0
0
0
1.12656410×10¹⁷
0
?
?
≥1
?
?
?
≥1
?
43
0
0
0
0
0
0
0
0
0
0
2.74152674×10¹⁷
0
?
?
?
?
?
?
?
?
44
0
2048
0
0
4194304
0
0
8.58993459×10¹⁰
0
0
6.84042446×10¹⁷
0
?
≥1
?
?
≥1
?
?
?
45
0
0
32768
0
0
0
1.07374182×10¹⁰
0
0
0
1.72750715×10¹⁸
0
?
?
≥1
?
?
?
≥1
?
46
0
0
0
0
0
0
0
0
0
0
4.25594852×10¹⁸
0
?
?
?
?
?
?
?
?
47
0
0
0
0
0
0
0
0
0
0
1.04907398×10¹⁹
0
?
?
?
?
?
?
?
?
48
0
4096
65536
0
16777216
6.87970713×10¹⁰
4.29496729×10¹⁰
6.87194767×10¹¹
2.78217462×10¹⁵
2.83360494×10¹⁶
2.63695495×10¹⁹
5.61434559×10¹⁹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
49
0
0
0
0
0
0
0
0
0
0
6.57172888×10¹⁹
0
?
?
?
?
?
?
?
?
50
0
0
0
0
0
0
0
0
0
0
1.61834879×10²⁰
0
?
?
?
?
?
?
?
?
51
0
0
131072
0
0
0
1.71798691×10¹¹
0
0
0
≥1.84467440×10²⁰
0
?
?
≥1
?
?
?
≥1
?
52
0
8192
0
0
67108864
0
0
5.49755813×10¹²
0
0
≥1.84467440×10²⁰
0
?
≥1
?
?
≥1
?
?
?
53
0
0
0
0
0
0
0
0
0
0
≥1.84467440×10²⁰
0
?
?
?
?
?
?
?
?
54
0
0
262144
0
0
0
6.87194767×10¹¹
0
0
0
≥1.84467440×10²⁰
0
?
?
≥1
?
?
?
≥1
?
55
0
0
0
0
0
0
0
0
0
0
≥1.84467440×10²⁰
0
?
?
?
?
?
?
?
?
56
0
16384
0
0
268435456
0
0
4.39804651×10¹³
0
0
≥1.84467440×10²⁰
0
?
≥1
?
?
≥1
?
?
?
57
0
0
524288
0
0
0
2.74877906×10¹²
0
0
0
≥1.84467440×10²⁰
0
?
?
≥1
?
?
?
≥1
?
58
0
0
0
0
0
0
0
0
0
0
≥1.84467440×10²⁰
0
?
?
?
?
?
?
?
?
59
0
0
0
0
0
0
0
0
0
0
≥1.84467440×10²⁰
0
?
?
?
?
?
?
?
?
60
0
32768
1048576
0
1.07374182×10¹⁰
1.98135565×10¹³
1.09951162×10¹³
3.51843720×10¹⁴
1.18825925×10¹⁹
≥1.84467440×10²⁰
≥1.84467440×10²⁰
≥1.84467440×10²⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
N>0
x
4k
3k
x
4k
12k
3k
4k
12k
12k
all
?
?
?
?
?
?
?
?

Smallest prime reptiles

Smallest prime reptile (6Fx8):

Reptile tilings' solutions count (including symmetric)

polyomino \ n²
F hexomino
1
0
0
0
0
0
0
≥3200000
≥1

Formulas

N(w;h) - number of ways to tile w×h rectangle (including symmetric solutions)

T(w;h)={1,N(w;h)10,else - tileability function, 1 if tiles rectangle, 0 otherwise

A(w;h)=(N(w;h))1wh - average number of ways to tile cell in w×h rectangle (including symmetric solutions)

G(T;x;y)=w=1h=1T(w;h)xwyh - bivariate generating function of T(w;h)

G(A;x;y)=w=1h=1A(w;h)xwyh - bivariate generating function of A(w;h)

N(1;n)=T(1;n)=0,n1

N(2;n)=T(2;n)=0,n1

N(3;n)=2×N(3;n4),n5

N(4;n)=2×N(4;n3),n4

N(5;n)=T(5;n)=0,n1

N(6;n)=4×N(6;n4),n5

N(7;n)=288×N(7;n12),n13

N(8;n)=4×N(8;n3),n4

N(9;n)=8×N(9;n4),n5

N(10;n)=4384×N(10;n12)483584×N(10;n24)+3604480×N(10;n36)+100663296×N(10;n48),n49

N(11;n)=7296×N(11;n12),n13

N(12;n)=8×N(12;n3)+20×N(12;n4)64×N(12;n8),n9

See Also

D hexominoI hexomino