# 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.

## 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):

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

## Formulas¶

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

$T(w; h) = \begin{cases} 1, & N(w; h) \geq 1 \\ 0, & \text{else} \end{cases}$ - tileability function, $1$ if tiles rectangle, $0$ otherwise

$A(w; h) = \left(N(w; h)\right)^{\frac{1}{wh}}$ - average number of ways to tile cell in $w\times h$ rectangle (including symmetric solutions)

$G(T; x; y) = \sum_{w=1}^{\infty}\sum _{h=1}^{\infty}T(w; h)x^wy^h$ - bivariate generating function of $T(w; h)$

$G(A; x; y) = \sum_{w=1}^{\infty}\sum _{h=1}^{\infty}A(w; h)x^wy^h$ - bivariate generating function of $A(w; h)$

$N(1; n) = T(1; n) = 0, \qquad n \geq 1 \tag{1}$

$N(2; n) = T(2; n) = 0, \qquad n \geq 1 \tag{2}$

$N(3; n) = 2 \times N(3; n - 4), \qquad n \geq 5 \tag{3}$

$N(4; n) = 2 \times N(4; n - 3), \qquad n \geq 4 \tag{4}$

$N(5; n) = T(5; n) = 0, \qquad n \geq 1 \tag{5}$

$N(6; n) = 4 \times N(6; n - 4), \qquad n \geq 5 \tag{6}$

$N(7; n) = 288 \times N(7; n - 12), \qquad n \geq 13 \tag{7}$

$N(8; n) = 4 \times N(8; n - 3), \qquad n \geq 4 \tag{8}$

$N(9; n) = 8 \times N(9; n - 4), \qquad n \geq 5 \tag{9}$

$N(10; n) = 4384 \times N(10; n - 12) - 483584 \times N(10; n - 24) + 3604480 \times N(10; n - 36) + 100663296 \times N(10; n - 48), \qquad n \geq 49 \tag{10}$

$N(11; n) = 7296 \times N(11; n - 12), \qquad n \geq 13 \tag{11}$

$N(12; n) = 8 \times N(12; n - 3) + 20 \times N(12; n - 4) - 64 \times N(12; n - 8), \qquad n \geq 9 \tag{12}$