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.


Y pentomino

Area: 5.

Perimeter: 12.

Size: 2x4.

Is rectangular: no.

Is convex: yes.

Holes: 0.

Order: 10.

Square order: 20.

Odd order: 45.

Prime rectangles: 40.

Smallest rectangle tilings

Smallest rectangle (5x10):

Smallest square (10x10):

Smallest odd rectangle (15x15):

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
1-4
5
6-8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
N>0
1-4
0
5
0
0
6-8
0
0
0
9
0
0
0
0
10
0
10
0
0
350
11
0
0
0
0
0
0
12
0
0
0
0
0
0
0
13
0
0
0
0
0
0
0
0
14
0
0
0
0
296
0
0
0
0
15
0
0
0
0
8954
0
0
0
9380
1696
16
0
0
0
0
50
0
0
0
0
20030
0
17
0
0
0
0
0
0
0
0
0
1152
0
0
18
0
0
0
0
0
0
0
0
0
0
0
0
0
19
0
0
0
0
13516
0
0
0
0
133880
0
0
0
0
20
0
100
0
224
242820
156
0
32
1107086
≥48554000
≥1
≥1
≥1
≥1
≥1
21
0
0
0
0
2760
0
0
0
0
≥1
0
0
0
0
≥1
0
22
0
0
0
0
0
0
0
0
0
≥1
0
0
0
0
≥1
0
0
23
0
0
0
0
5832
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
24
0
0
0
0
602980
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
25
0
0
0
0
6591140
0
0
0
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
all
26
0
0
0
0
126088
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
27
0
0
0
0
496
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
28
0
0
0
0
442978
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
29
0
0
0
0
22506998
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
30
0
1000
0
12420
179340480
23992
0
≥68000
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
all
31
0
0
0
0
4843754
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
32
0
0
0
0
147924
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
33
0
0
0
0
24885442
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
34
0
0
0
0
789989882
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
35
0
0
0
0
4.89108913×10¹⁰
48504
0
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
all
36
0
0
0
0
172488058
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
37
0
0
0
0
13913138
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
38
0
0
0
0
1.18457646×10¹⁰
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
39
0
0
0
0
2.63870658×10¹¹
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
40
0
10000
0
579456
1.33548940×10¹²
10951530
0
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
all
41
0
0
0
0
5.82375181×10¹⁰
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
42
0
0
0
0
928325010
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
43
0
0
0
0
5.05154848×10¹¹
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
44
0
0
0
0
8.53127100×10¹²
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
45
0
0
0
640
3.65025521×10¹³
75341532
0
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
all
46
0
0
0
0
1.89958771×10¹²
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
47
0
0
0
0
5.17668000×10¹¹
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
48
0
0
0
0
2.00310134×10¹³
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
49
0
0
0
0
2.69301190×10¹⁴
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
50
0
100000
0
27532292
9.98522740×10¹⁴
7.88568587×10¹⁰
65536
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
all
51
0
0
0
0
6.04698126×10¹³
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
52
0
0
0
0
2.55608729×10¹³
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
53
0
0
0
0
7.52541067×10¹⁴
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
54
0
0
0
0
8.34898388×10¹⁵
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
55
0
0
0
78096
2.73343192×10¹⁶
6.27623714×10¹¹
2002400
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
all
56
0
0
0
0
1.89242891×10¹⁵
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
57
0
0
0
0
1.15762741×10¹⁵
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
58
0
0
0
0
2.71423891×10¹⁶
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
59
0
0
0
0
2.55239746×10¹⁷
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
60
0
1000000
0
1.33167840×10¹⁰
7.48774482×10¹⁷
6.07504380×10¹³
56922624
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
all
61
0
0
0
0
5.85274226×10¹⁶
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
62
0
0
0
0
4.91212107×10¹⁶
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
63
0
0
0
0
9.48163614×10¹⁷
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
64
0
0
0
0
7.71652217×10¹⁸
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
65
0
0
0
7176128
2.05246057×10¹⁹
4.68636448×10¹⁴
1.64902008×10¹⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
all
66
0
0
0
0
1.79593964×10¹⁸
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
67
0
0
0
0
1.98089015×10¹⁸
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
68
0
0
0
0
3.22847275×10¹⁹
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
69
0
0
0
0
2.31186408×10²⁰
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
70
0
10000000
0
6.49366215×10¹¹
5.62956329×10²⁰
4.60042342×10¹⁶
5.16585390×10¹¹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
all
71
0
0
0
0
5.48506546×10¹⁹
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
72
0
0
0
0
7.66874511×10¹⁹
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
73
0
0
0
0
1.07652404×10²¹
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
74
0
0
0
0
6.87469458×10²¹
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
75
0
0
0
551013048
1.54507809×10²²
3.43980326×10¹⁷
1.72883751×10¹³
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
all
76
0
0
0
0
1.67155350×10²¹
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
77
0
0
0
0
2.87124858×10²¹
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
78
0
0
0
0
3.52780553×10²²
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
79
0
0
0
0
2.03151826×10²³
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
80
0
100000000
0
3.18081201×10¹³
4.24331934×10²³
3.41495393×10¹⁹
6.07912516×10¹⁴
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
all
81
0
0
0
0
5.09296950×10²²
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
82
0
0
0
0
1.04551400×10²³
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
83
0
0
0
0
1.13928605×10²⁴
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
84
0
0
0
0
5.97141363×10²⁴
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
85
0
0
0
3.81474728×10¹¹
1.16613287×10²⁵
2.52196392×10²⁰
2.23890008×10¹⁶
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
all
86
0
0
0
0
1.55380398×10²⁴
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
87
0
0
0
0
3.71863273×10²⁴
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
88
0
0
0
0
3.63369444×10²⁵
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
89
0
0
0
0
1.74724086×10²⁶
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
90
0
1.00000000×10¹⁰
0
1.56281224×10¹⁵
3.20692506×10²⁶
2.50728914×10²²
8.59316629×10¹⁷
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
all
91
0
0
0
0
4.75194067×10²⁵
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
92
0
0
0
0
1.29633608×10²⁶
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
93
0
0
0
0
1.14658024×10²⁷
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
94
0
0
0
0
5.09232071×10²⁷
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
95
0
0
0
2.49089530×10¹³
8.82558826×10²⁷
1.85163999×10²³
3.42123297×10¹⁹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
all
96
0
0
0
0
1.45779854×10²⁷
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
97
0
0
0
0
4.44146681×10²⁷
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
98
0
0
0
0
3.58436604×10²⁸
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
99
0
0
0
0
1.47906427×10²⁹
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
5k
100
0
1.00000000×10¹¹
0
7.69538866×10¹⁶
2.43070780×10²⁹
1.83199211×10²⁵
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
all
N>0
x
10k
x
5k
all
5k
5k
5k
5k
all
5k
5k
5k
5k
all
5k
5k
5k

Smallest prime reptiles

Smallest prime reptile (5Yx9):

Reptile tilings' solutions count (including symmetric)

polyomino \ n²
10²
11²
12²
13²
Y pentomino
1
0
0
0
0
0
0
0
7629
≥46742129182700
≥20000
≥512
≥262144

Smallest tori tilings

Smallest torus (2x5):

Smallest square torus (5x5):

Smallest odd torus (5x5):

Tori tilings' solutions count (including translations)

w \ h
1
2
3
4
5
6
7
8
1
0
2
0
0
3
0
0
0
4
0
0
0
0
5
0
40
0
260
40
6
0
0
0
0
2320
0
7
0
0
0
0
140
0
0
8
0
0
0
0
≥10000
0
0
0
9
0
0
0
0
720
0
0
0
10
0
160
120
1420
≥10000
≥10000
≥10000
≥10000
11
0
0
0
0
3300
0
0
0
12
0
0
0
0
≥10000
0
0
0
13
0
0
0
0
≥10000
0
0
0
14
0
0
0
0
≥10000
0
0
0
15
0
640
0
≥10000
≥10000
≥10000
≥10000
≥10000
16
0
0
0
0
≥10000
0
0
0
17
0
0
0
0
≥10000
0
0
0
18
0
0
0
0
≥10000
0
0
0
19
0
0
0
0
≥10000
0
0
0
20
0
2560
480
≥10000
≥10000
≥10000
≥10000
≥10000
21
0
0
0
0
≥10000
0
0
0
22
0
0
0
0
≥10000
0
0
0
23
0
0
0
0
≥10000
0
0
0
24
0
0
0
0
≥10000
0
0
0
25
0
≥10000
0
≥10000
≥10000
≥10000
≥10000
≥10000
26
0
0
0
0
≥10000
0
0
0
27
0
0
0
0
≥10000
0
0
0
28
0
0
0
0
≥10000
0
0
0
29
0
0
0
0
≥10000
0
0
0
30
0
≥10000
1920
≥10000
≥10000
≥10000
≥10000
?

Smallest Baiocchi figures

Smallest Baiocchi figure (area 20):

Smallest known Baiocchi figure without holes (area 40):

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 $A(w; h)$

$N(1; n) = T(1; n) = A(1; n) = 0 \tag{1}$

$N(2; n) = T(2; n) = A(2; n) = 0 \tag{2}$

$N(3; n) = T(3; n) = A(3; n) = 0 \tag{3}$

$N(4; n) = T(4; n) = A(4; n) = 0 \tag{4}$

$N(5; n) = 10 \times N(5; n - 10), \quad n \geq 11 \tag{5}$

$N(6; n) = T(6; n) = A(6; n) = 0 \tag{6}$

$N(7; n) = T(7; n) = A(7; n) = 0 \tag{7}$

$N(8; n) = T(8; n) = A(8; n) = 0 \tag{8}$

$G(N(5); x) = \frac{1}{1 - 10x^{10}} \tag{9}$

Attributions

  1. Smallest rectangle and smallest odd rectangle taken from http://www.cflmath.com/~reid/Polyomino/y5_rect.html
  2. Prime rectangles taken from http://www2.stetson.edu/~efriedma/order/index.html

See Also

X pentominoZ pentomino