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.


Monominoes and Dominoes

Prime rectangles: 4.

Smallest rectangle tilings

Smallest rectangle (1x3):

Smallest square (2x2):

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
0
2
0
4
3
2
18
130
4
3
65
811
9975
5
7
219
5095
120369
≥1
6
11
719
31645
1453253
≥1
≥1
7
20
2334
196784
17524837
≥1
≥1
≥1
8
32
7538
1222396
211349699
≥1
≥1
≥1
≥1
9
54
24286
7594360
2.54867831×10¹⁰
≥1
≥1
≥1
≥1
≥1
10
87
78153
47176525
3.07349144×10¹¹
≥1
≥1
≥1
≥1
≥1
≥1
11
143
251353
293066687
3.70635173×10¹²
≥1
≥1
≥1
≥1
≥1
≥1
≥1
12
231
808161
1.82055016×10¹⁰
4.46952717×10¹³
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
13
376
2598062
1.13093959×10¹¹
5.38984611×10¹⁴
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
14
608
8351606
7.02547597×10¹¹
6.49966806×10¹⁵
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
15
986
26845708
4.36427542×10¹²
7.83801297×10¹⁶
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
16
1595
86292267
2.71111854×10¹³
9.45193618×10¹⁷
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
17
2583
277373489
1.68416589×10¹⁴
1.13981818×10¹⁹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
18
4179
891571209
1.04621568×10¹⁵
1.37451784×10²⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
19
6764
2.86580161×10¹⁰
6.49916534×10¹⁵
1.65754445×10²¹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
20
10944
9.21161351×10¹⁰
4.03732717×10¹⁶
1.99884899×10²²
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
21
17710
2.96090886×10¹¹
2.50801600×10¹⁷
2.41043146×10²³
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
22
28655
9.51731065×10¹¹
1.55799715×10¹⁸
2.90676277×10²⁴
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
23
46367
3.05916841×10¹²
9.67838780×10¹⁸
3.50529353×10²⁵
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
24
75023
9.83314616×10¹²
6.01228249×10¹⁹
4.22706760×10²⁶
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
25
121392
3.16068770×10¹³
3.73487212×10²⁰
5.09746197×10²⁷
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
26
196416
1.01594610×10¹⁴
2.32012879×10²¹
6.14707902×10²⁸
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
27
317810
3.26557566×10¹⁴
1.44128030×10²²
7.41282244×10²⁹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
28
514227
1.04966043×10¹⁵
8.95333452×10²²
8.93919476×10³⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
29
832039
3.37394428×10¹⁵
5.56187430×10²³
1.07798620×10³²
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
30
1346267
1.08449357×10¹⁶
3.45507539×10²⁴
1.29995405×10³³
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
31
2178308
3.48590910×10¹⁶
2.14631710×10²⁵
1.56762725×10³⁴
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
32
3524576
1.12048264×10¹⁷
1.33330726×10²⁶
1.89041696×10³⁵
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
33
5702886
3.60158949×10¹⁷
8.28259838×10²⁶
2.27967222×10³⁶
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
34
9227463
1.15766602×10¹⁸
5.14520830×10²⁷
2.74907893×10³⁷
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
35
14930351
3.72110875×10¹⁸
3.19623954×10²⁸
3.31514104×10³⁸
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
36
24157815
1.19608333×10¹⁹
1.98552645×10²⁹
3.99776086×10³⁹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
37
39088168
3.84459426×10¹⁹
1.23342298×10³⁰
4.82093875×10⁴⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
38
63245984
1.23577552×10²⁰
7.66211022×10³⁰
5.81361698×10⁴¹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
39
102334154
3.97217767×10²⁰
4.75975669×10³¹
7.01069731×10⁴²
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
40
165580139
1.27678491×10²¹
2.95679429×10³²
8.45426815×10⁴³
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
41
267914295
4.10399495×10²¹
1.83678138×10³³
1.01950842×10⁴⁵
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
42
433494435
1.31915519×10²²
1.14102150×10³⁴
1.22943513×10⁴⁶
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
43
701408732
4.24018660×10²²
7.08810573×10³⁴
1.48258780×10⁴⁷
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
44
1.13490316×10¹⁰
1.36293155×10²³
4.40318107×10³⁵
1.78786706×10⁴⁸
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
45
1.83631190×10¹⁰
4.38089779×10²³
2.73528701×10³⁶
2.15600630×10⁴⁹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
46
2.97121507×10¹⁰
1.40816062×10²⁴
1.69917950×10³⁷
2.59994899×10⁵⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
47
4.80752697×10¹⁰
4.52627850×10²⁴
1.05554224×10³⁸
3.13530380×10⁵¹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
48
7.77874204×10¹⁰
1.45489063×10²⁵
6.55710259×10³⁸
3.78089338×10⁵²
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
49
1.25862690×10¹¹
4.67648369×10²⁵
4.07331819×10³⁹
4.55941615×10⁵³
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
50
2.03650110×10¹¹
1.50317138×10²⁶
2.53037387×10⁴⁰
5.49824432×10⁵⁴
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
51
3.29512800×10¹¹
4.83167347×10²⁶
1.57188602×10⁴¹
6.63038636×10⁵⁵
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
52
5.33162911×10¹¹
1.55305434×10²⁷
9.76466644×10⁴¹
7.99564747×10⁵⁶
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
53
8.62675712×10¹¹
4.99201323×10²⁷
6.06587939×10⁴²
9.64202913×10⁵⁷
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
54
1.39583862×10¹²
1.60459267×10²⁸
3.76816689×10⁴³
1.16274168×10⁵⁹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
55
2.25851433×10¹²
5.15767390×10²⁸
2.34081174×10⁴⁴
1.40216151×10⁶⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
56
3.65435296×10¹²
1.65784130×10²⁹
1.45412869×10⁴⁵
1.69088022×10⁶¹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
57
5.91286729×10¹²
5.32883203×10²⁹
9.03314963×10⁴⁵
2.03904892×10⁶²
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
58
9.56722026×10¹²
1.71285700×10³⁰
5.61145601×10⁴⁶
2.45890894×10⁶³
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
59
1.54800875×10¹³
5.50567008×10³⁰
3.48587589×10⁴⁷
2.96522224×10⁶⁴
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
60
2.50473078×10¹³
1.76969840×10³¹
2.16545059×10⁴⁸
3.57579037×10⁶⁵
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
61
4.05273953×10¹³
5.68837652×10³¹
1.34519312×10⁴⁹
4.31208042×10⁶⁶
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
62
6.55747031×10¹³
1.82842609×10³²
8.35643416×10⁴⁹
5.19997976×10⁶⁷
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
63
1.06102098×10¹⁴
5.87714610×10³²
5.19107561×10⁵⁰
6.27070623×10⁶⁸
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
64
1.71676801×10¹⁴
1.88910267×10³³
3.22473263×10⁵¹
7.56190569×10⁶⁹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
65
2.77778900×10¹⁴
6.07218002×10³³
2.00322656×10⁵²
9.11897570×10⁷⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
66
4.49455702×10¹⁴
1.95179281×10³⁴
1.24441841×10⁵³
1.09966615×10⁷²
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
67
7.27234602×10¹⁴
6.27368617×10³⁴
7.73041462×10⁵³
1.32609812×10⁷³
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
68
1.17669030×10¹⁵
2.01656333×10³⁵
4.80218787×10⁵⁴
1.59915464×10⁷⁴
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
69
1.90392490×10¹⁵
6.48187933×10³⁵
2.98315284×10⁵⁵
1.92843615×10⁷⁵
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
70
3.08061521×10¹⁵
2.08348327×10³⁶
1.85315550×10⁵⁶
2.32551993×10⁷⁶
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
71
4.98454011×10¹⁵
6.69698141×10³⁶
1.15119321×10⁵⁷
2.80436713×10⁷⁷
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
72
8.06515533×10¹⁵
2.15262395×10³⁷
7.15129312×10⁵⁷
3.38181362×10⁷⁸
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
73
1.30496954×10¹⁶
6.91922169×10³⁷
4.44243352×10⁵⁸
4.07816198×10⁷⁹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
74
2.11148507×10¹⁶
2.22405909×10³⁸
2.75967091×10⁵⁹
4.91789523×10⁸⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
75
3.41645462×10¹⁶
7.14883704×10³⁸
1.71432695×10⁶⁰
5.93053775×10⁸¹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
76
5.52793970×10¹⁶
2.29786480×10³⁹
1.06495195×10⁶¹
7.15169321×10⁸²
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
77
8.94439432×10¹⁶
7.38607221×10³⁹
6.61555633×10⁶¹
8.62429647×10⁸³
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
78
1.44723340×10¹⁷
2.37411977×10⁴⁰
4.10963006×10⁶²
1.04001230×10⁸⁵
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
79
2.34167283×10¹⁷
7.63118006×10⁴⁰
2.55293106×10⁶³
1.25416096×10⁸⁶
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
80
3.78890623×10¹⁷
2.45290527×10⁴¹
1.58589871×10⁶⁴
1.51240492×10⁸⁷
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
81
6.13057907×10¹⁷
7.88442184×10⁴¹
9.85171421×10⁶⁴
1.82382382×10⁸⁸
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
82
9.91948530×10¹⁷
2.53430528×10⁴²
6.11995408×10⁶⁵
2.19936690×10⁸⁹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
83
1.60500643×10¹⁸
8.14606750×10⁴²
3.80175846×10⁶⁶
2.65223796×10⁹⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
84
2.59695496×10¹⁸
2.61840656×10⁴³
2.36167906×10⁶⁷
3.19835958×10⁹¹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
85
4.20196140×10¹⁸
8.41639590×10⁴³
1.46709162×10⁶⁸
3.85693296×10⁹²
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
86
6.79891637×10¹⁸
2.70529875×10⁴⁴
9.11367625×10⁶⁸
4.65111301×10⁹³
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
87
1.10008777×10¹⁹
8.69569518×10⁴⁴
5.66147971×10⁶⁹
5.60882246×10⁹⁴
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
88
1.77997941×10¹⁹
2.79507447×10⁴⁵
3.51695097×10⁷⁰
6.76373362×10⁹⁵
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
89
2.88006719×10¹⁹
8.98426306×10⁴⁵
2.18475464×10⁷¹
8.15645223×10⁹⁶
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
90
4.66004661×10¹⁹
2.88782941×10⁴⁶
1.35718493×10⁷²
9.83594516×10⁹⁷
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
91
7.54011380×10¹⁹
9.28240710×10⁴⁶
8.43092816×10⁷²
1.18612620×10⁹⁹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
92
1.22001604×10²⁰
2.98366244×10⁴⁷
5.23735180×10⁷³
1.43036113×10¹⁰⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
93
1.97402742×10²⁰
9.59044509×10⁴⁷
3.25347973×10⁷⁴
1.72488642×10¹⁰¹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
94
3.19404346×10²⁰
3.08267570×10⁴⁸
2.02108446×10⁷⁵
2.08005733×10¹⁰²
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
95
5.16807088×10²⁰
9.90870536×10⁴⁸
1.25551186×10⁷⁶
2.50836138×10¹⁰³
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
96
8.36211434×10²⁰
3.18497472×10⁴⁹
7.79932792×10⁷⁶
3.02485741×10¹⁰⁴
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
97
1.35301852×10²¹
1.02375271×10⁵⁰
4.84499729×10⁷⁷
3.64770501×10¹⁰⁵
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
98
2.18922995×10²¹
3.29066856×10⁵⁰
3.00974635×10⁷⁸
4.39880298×10¹⁰⁶
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
99
3.54224848×10²¹
1.05772609×10⁵¹
1.86967558×10⁷⁹
5.30455935×10¹⁰⁷
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
100
5.73147844×10²¹
3.39986986×10⁵¹
1.16145560×10⁸⁰
6.39681976×10¹⁰⁸
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
N>0
all
all
all
all
all
all
all
all
all
?
?
?
?
?
?
?
?
?
?
?

Smallest prime reptiles

Smallest prime reptiles (1Ox2, 2Ix2):

Reptile tilings' solutions count (including symmetric)

polyomino \ n²
O monomino
?
?
?
?
I domino
?
?
?
?

Smallest common multiples

Smallest common multiple (area 2):

Common multiples' solutions count (excluding symmetric)

area
2
solutions
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 $A(w; h)$

$G(N(1); x) = \frac{2x^3 + x^4}{1 - x - 2x^2 + x^3 + x^4} \tag{1}$

$G(N(2); x) = \frac{4x^2 - 2x^3 - 5x^4 + x^6}{1 - 5x + 5x^2 + 4x^3 - 5x^4 - x^5 + x^6} \tag{2}$

$G(N(3); x) = \frac{2x + 8x^2 + 12x^3 - 23x^4 - 66x^5 + 12x^6 + 53x^7 - x^8 - 14x^9 + x^{11}}{1 - 5x - 14x^2 + 34x^3 + 51x^4 - 71x^5 - 51x^6 + 55x^7 + 14x^8 - 14x^9 - x^{10} + x^{11}} \tag{3}$

$G(N(4); x) = \frac{3x + 32x^2 + 15x^3 - 212x^4 - 181x^5 + 417x^6 + 465x^7 - 391x^8 - 315x^9 + 167x^{10} + 58x^{11} - 27x^{12} - 2x^{13} + x^{14}}{1 - 11x - 27x^2 + 163x^3 + 159x^4 - 660x^5 - 200x^6 + 1012x^7 - 112x^8 - 504x^9 + 123x^{10} + 83x^{11} - 25x^{12} - 3x^{13} + x^{14}} \tag{4}$

See Also

P3 9-ominoMonominoes and I triomino