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.


U pentomino and C hexomino

Prime rectangles: ≥ 89.

Smallest rectangle tilings

Smallest rectangle (12x14):

Smallest known square (22x22):

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
1-11
12
13
14
15-16
17
18
19
20
21
22
23
N>0
1-11
0
12
0
0
13
0
0
0
14
0
1
0
0
15-16
0
0
0
0
0
17
0
0
0
0
0
0
18
0
1
0
0
0
0
0
19
0
0
0
0
0
0
0
0
20
0
0
0
0
0
0
2
0
0
21
0
0
0
0
0
0
0
0
0
0
22
0
1
0
0
0
0
0
0
0
0
8
23
0
4
0
0
0
0
0
0
0
0
?
0
24
0
0
0
2
0
0
2
0
≥1
?
≥1
≥1
?
25
0
0
0
0
0
0
0
0
0
0
?
0
?
26
0
1
0
0
0
0
4
0
≥1
?
?
?
?
27
0
16
0
0
0
0
0
0
0
0
?
0
?
28
0
1
0
0
0
0
12
0
0
?
?
?
?
29
0
2
0
0
0
0
0
0
0
0
?
0
?
30
0
1
0
0
0
0
18
0
≥1
?
?
?
?
31
0
36
0
0
0
0
0
0
0
0
?
0
?
32
0
11
0
2
0
0
11
0
≥1
?
?
?
?
33
0
4
0
0
0
0
0
0
≥1
0
?
0
?
34
0
1
0
0
0
0
110
0
≥1
?
≥1
?
?
35
0
64
0
0
0
0
4
0
≥1
0
?
0
?
36
0
106
0
4
0
0
114
0
≥1
?
≥1
≥1
?
37
0
14
0
0
0
0
12
0
≥1
0
?
0
?
38
0
21
0
0
0
0
298
0
?
?
?
?
?
39
0
100
0
0
0
0
120
0
?
0
?
0
?
40
0
451
0
0
0
4
684
0
?
?
?
?
?
41
0
64
0
0
0
0
120
0
?
0
?
0
?
42
0
94
0
0
0
4
1386
0
≥1
?
?
?
?
43
0
148
0
0
0
0
928
0
?
0
?
0
?
44
0
1316
0
8
0
0
3271
≥1
≥1
?
≥1
?
?
45
0
562
0
0
0
0
1682
0
?
0
?
0
?
46
0
288
0
0
0
4
7580
0
?
?
≥1
?
?
47
0
324
0
0
0
0
5786
0
?
0
?
0
?
48
0
3061
8
8
0
6
16948
?
≥1
?
≥1
≥1
?
49
0
3560
0
0
0
0
15628
0
?
0
?
0
?
50
0
920
0
0
0
4
42840
?
≥1
?
?
?
?
51
0
1256
0
0
0
0
41020
0
≥1
0
?
0
?
52
0
6198
0
4
0
16
101933
?
≥1
?
?
?
?
53
0
15386
0
0
0
0
113044
0
≥1
0
?
0
?
54
0
4206
0
0
0
24
247386
?
≥1
?
?
?
?
55
0
4700
0
0
0
0
301828
0
≥1
0
?
0
?
56
0
11938
16
24
0
80
651122
?
≥1
?
≥1
?
?
57
0
50534
0
0
0
0
794424
0
≥1
0
?
0
?
58
0
25273
4
0
0
70
1589792
?
≥1
?
≥1
?
?
59
0
15872
0
0
0
0
2175112
0
≥1
0
?
0
?
60
0
26537
32
16
0
114
4258736
?
≥1
?
≥1
≥1
?
61
0
137214
0
0
0
0
5619550
0
≥1
0
?
0
?
62
0
135539
0
0
0
172
10904014
?
≥1
?
?
?
?
63
0
55588
0
0
0
0
15185058
0
≥1
0
?
0
?
64
0
76517
32
24
0
388
28912980
?
≥1
?
?
?
?
65
0
327542
0
0
0
0
40297620
0
≥1
0
?
0
?
66
0
580298
20
0
0
426
75597026
?
≥1
?
≥1
?
?
67
0
236052
0
0
0
0
106215428
0
≥1
0
?
0
?
68
0
253935
152
64
0
1018
202889134
?
≥1
?
≥1
?
?
69
0
738178
0
0
0
0
288516052
0
≥1
0
?
0
?
70
0
2031699
8
0
0
1072
530656520
?
≥1
?
≥1
?
?
71
0
1171300
0
0
0
0
758175234
0
≥1
0
?
0
?
72
0
877318
140
42
0
2832
1.43020650×10¹⁰
?
≥1
?
≥1
≥1
?
73
0
1743440
0
0
0
0
2.06875259×10¹⁰
0
≥1
0
?
0
?
74
0
6079350
68
0
0
3398
3.77034849×10¹⁰
?
≥1
?
?
?
?
75
0
5621328
0
0
0
0
5.47839852×10¹⁰
0
≥1
0
?
0
?
76
0
3182868
456
96
0
7168
1.00706953×10¹¹
?
≥1
?
?
?
?
77
0
4718252
0
0
0
0
1.49217365×10¹¹
0
≥1
0
?
0
?
78
0
16250778
104
0
0
11242
2.68709717×10¹¹
?
≥1
?
≥1
?
?
79
0
23573226
0
0
0
0
3.97723568×10¹¹
0
≥1
0
?
0
?
80
0
12841202
760
160
0
21242
7.12745243×10¹¹
?
≥1
?
≥1
?
?
81
0
14705818
0
0
0
0
1.08182301×10¹²
0
≥1
0
?
0
?
82
0
40944437
224
8
0
33594
1.91320059×10¹²
?
≥1
?
≥1
?
?
83
0
85381128
0
0
0
0
2.89521398×10¹²
0
≥1
0
?
0
?
84
0
56852904
1390
140
0
55006
5.07501754×10¹²
?
≥1
?
≥1
≥1
?
85
0
50100126
0
0
0
0
7.85052471×10¹²
0
≥1
0
?
0
?
86
0
104167143
588
0
0
94142
1.36464341×10¹³
?
≥1
?
?
?
?
87
0
272548976
0
0
0
0
2.10873626×10¹³
0
≥1
0
?
0
?
88
0
250904667
3234
340
0
168600
3.62814999×10¹³
≥1
≥1
?
≥1
?
?
89
0
181348338
0
0
0
0
5.69240074×10¹³
0
≥1
0
?
0
?
90
0
286457116
904
8
0
274240
9.76692057×10¹³
?
≥1
?
≥1
?
?
91
0
790950544
0
0
0
0
1.53362559×10¹⁴
0
≥1
0
?
0
?
92
0
1.02331150×10¹⁰
5016
418
0
508818
2.60325831×10¹⁴
?
≥1
?
≥1
?
?
93
0
703455004
0
0
0
0
4.12533276×10¹⁴
0
≥1
0
?
0
?
94
0
877291954
2520
60
0
876152
7.01115314×10¹⁴
?
≥1
?
≥1
?
?
95
0
2.17295971×10¹⁰
0
0
0
0
1.11303720×10¹⁵
0
≥1
0
?
0
?
96
0
3.76338141×10¹⁰
11864
496
0
1576126
1.87399292×10¹⁵
?
≥1
?
≥1
≥1
?
97
0
2.89123858×10¹⁰
0
0
0
0
2.98866378×10¹⁵
0
≥1
0
?
0
?
98
0
2.93670831×10¹⁰
4412
16
0
2841214
5.04408753×10¹⁵
?
≥1
?
?
?
?
99
0
5.94263198×10¹⁰
0
0
0
0
8.06669000×10¹⁵
0
≥1
0
?
0
?
100
0
1.25496555×10¹¹
20508
1064
0
4986842
1.35183938×10¹⁶
?
≥1
?
≥1
?
?
101
0
1.19628223×10¹¹
0
0
0
0
2.16483947×10¹⁶
0
≥1
0
?
0
?
102
0
1.04657857×10¹¹
10204
136
0
9115236
3.63488589×10¹⁶
?
≥1
?
≥1
?
?
103
0
1.70202133×10¹¹
0
0
0
0
5.84250828×10¹⁶
0
≥1
0
?
0
?
104
0
3.87297036×10¹¹
42204
1202
0
16373768
9.76145466×10¹⁶
?
≥1
?
≥1
?
?
105
0
4.73443823×10¹¹
0
0
0
0
1.56837979×10¹⁷
0
≥1
0
?
0
?
106
0
3.93009127×10¹¹
21120
252
0
29194074
2.62237006×10¹⁷
?
≥1
?
≥1
?
?
107
0
5.25634922×10¹¹
0
0
0
0
4.23079456×10¹⁷
0
≥1
0
?
0
?
108
0
1.14011149×10¹²
84226
1942
0
53602916
7.05142440×10¹⁷
?
≥1
?
≥1
≥1
?
109
0
1.74706236×10¹²
0
0
0
0
1.13659960×10¹⁸
0
≥1
0
?
0
?
110
0
1.53452459×10¹²
43924
204
0
96160852
1.89333267×10¹⁸
?
≥1
?
≥1
?
?
111
0
1.74541271×10¹²
0
0
0
0
3.06387312×10¹⁸
0
≥1
0
?
0
?
112
0
3.32331623×10¹²
159316
3280
0
176140388
5.09468898×10¹⁸
?
≥1
?
≥1
?
?
113
0
5.99587410×10¹²
0
0
0
0
8.23827178×10¹⁸
0
≥1
0
?
0
?
114
0
6.04701724×10¹²
98412
986
0
319798472
1.36777381×10¹⁹
?
≥1
?
≥1
?
?
115
0
6.12705298×10¹²
0
0
0
0
2.21909037×10¹⁹
0
≥1
0
?
0
?
116
0
9.95368746×10¹²
337628
3868
0
586865206
3.68154210×10¹⁹
?
≥1
?
≥1
?
?
117
0
1.93732866×10¹³
0
0
0
0
5.97109198×10¹⁹
0
≥1
0
?
0
?
118
0
2.32643509×10¹³
206548
1344
0
1.07586072×10¹⁰
9.88616803×10¹⁹
?
≥1
?
≥1
?
?
119
0
2.24277544×10¹³
0
0
0
0
1.60735934×10²⁰
0
≥1
0
?
0
?
120
0
3.13607496×10¹³
652962
7224
0
1.97614858×10¹⁰
2.66099958×10²⁰
?
≥1
?
≥1
≥1
?
121
0
6.02269319×10¹³
0
0
0
0
4.32712365×10²⁰
0
≥1
0
?
0
?
122
0
8.55433059×10¹³
472116
2062
0
3.63400238×10¹⁰
7.14887959×10²⁰
?
≥1
?
≥1
?
?
123
0
8.44723502×10¹³
0
0
0
0
1.16427960×10²¹
0
≥1
0
?
0
?
124
0
1.04276570×10¹⁴
1383568
11118
0
6.70794192×10¹⁰
1.92393731×10²¹
?
≥1
?
≥1
?
?
125
0
1.85108670×10¹⁴
0
0
0
0
3.13506381×10²¹
0
≥1
0
?
0
?
126
0
2.98846121×10¹⁴
1044372
5098
0
1.23445869×10¹¹
5.17135226×10²¹
?
≥1
?
≥1
?
?
127
0
3.21140525×10¹⁴
0
24
0
0
8.43324752×10²¹
0
≥1
0
?
0
?
128
0
3.62539034×10¹⁴
2855316
15222
0
2.28484013×10¹¹
1.39146193×10²²
?
≥1
?
≥1
?
?
129
0
5.77583080×10¹⁴
0
32
0
0
2.27091951×10²²
0
≥1
0
?
0
?
130
0
9.98210345×10¹⁴
2422288
7944
0
4.22029951×10¹¹
3.74182032×10²²
?
≥1
?
≥1
?
?
131
0
1.20619217×10¹⁵
0
16
0
0
6.10833434×10²²
0
≥1
0
?
0
?
132
0
1.30302547×10¹⁵
6086320
27120
0
7.81337135×10¹¹
1.00663059×10²³
≥1
≥1
?
≥1
≥1
?
133
0
1.86238374×10¹⁵
0
40
0
0
1.64471896×10²³
0
≥1
0
?
0
?
134
0
3.23505308×10¹⁵
5635712
14668
0
1.44825098×10¹²
2.70790914×10²³
?
≥1
?
≥1
?
?
135
0
4.40521382×10¹⁵
0
80
0
0
4.42426613×10²³
0
≥1
0
?
0
?
136
0
4.78687799×10¹⁵
13319174
41066
0
2.68313125×10¹²
7.28392148×10²³
?
≥1
?
≥1
?
?
137
0
6.23675814×10¹⁵
0
88
0
0
1.19108992×10²⁴
0
≥1
0
?
0
?
138
0
1.03682196×10¹⁶
13408428
29082
0
4.98092985×10¹²
1.95987299×10²⁴
?
≥1
?
≥1
?
?
139
0
1.55429321×10¹⁶
0
160
0
0
3.20442312×10²⁴
0
≥1
0
?
0
?
140
0
1.77385604×10¹⁶
29418634
64904
0
9.24966125×10¹²
5.27151607×10²⁴
?
≥1
?
≥1
?
?
141
0
2.15959496×10¹⁶
0
260
0
0
8.62542898×10²⁴
0
≥1
0
?
0
?
142
0
3.34780795×10¹⁶
32368680
47106
0
1.71782215×10¹³
1.41856328×10²⁵
?
≥1
?
≥1
?
?
143
0
5.31161976×10¹⁶
0
348
0
0
2.32084623×10²⁵
0
≥1
0
?
0
?
144
0
6.53805533×10¹⁶
67260536
117276
0
3.19493415×10¹³
3.81560150×10²⁵
?
≥1
?
≥1
≥1
?
145
0
7.67154466×10¹⁶
0
436
0
0
6.24614065×10²⁵
0
≥1
0
?
0
?
146
0
1.10335137×10¹⁷
79164124
86662
0
5.94137339×10¹³
1.02681283×10²⁶
?
≥1
?
≥1
?
?
147
0
1.77444393×10¹⁷
0
802
0
0
1.68084362×10²⁶
0
≥1
0
?
0
?
148
0
2.37001167×10¹⁷
155906188
179100
0
1.10563483×10¹⁴
2.76206300×10²⁶
?
≥1
?
≥1
?
?
149
0
2.77169560×10¹⁷
0
1368
0
0
4.52316934×10²⁶
0
≥1
0
?
0
?
150
0
3.73020616×10¹⁷
196133936
170236
0
2.05828982×10¹⁴
7.43284890×10²⁶
?
≥1
?
≥1
?
?
151
0
5.86937413×10¹⁷
0
1670
0
0
1.21728015×10²⁷
0
≥1
0
?
0
?
152
0
8.40158859×10¹⁷
369736262
300982
0
3.83270999×10¹⁴
1.99956856×10²⁷
?
≥1
?
≥1
?
?
153
0
1.00912390×10¹⁸
0
2304
0
0
3.27548085×10²⁷
0
≥1
0
?
0
?
154
0
1.29159585×10¹⁸
490211616
287002
0
7.14067330×10¹⁴
5.38073107×10²⁷
?
≥1
?
≥1
?
?
155
0
1.94689888×10¹⁸
0
3718
0
0
8.81529177×10²⁷
0
≥1
0
?
0
?
156
0
2.91396329×10¹⁸
892908366
540836
0
1.33043710×10¹⁵
1.44764575×10²⁸
?
≥1
?
≥1
≥1
?
157
0
3.66785650×10¹⁸
0
5240
0
0
2.37196121×10²⁸
0
≥1
0
?
0
?
158
0
4.55784785×10¹⁸
1.23645671×10¹⁰
534398
0
2.47983153×10¹⁵
3.89537158×10²⁸
?
≥1
?
≥1
?
?
159
0
6.53709000×10¹⁸
0
7306
0
0
6.38364388×10²⁸
0
≥1
0
?
0
?
160
0
9.94182144×10¹⁸
2.18489976×10¹⁰
890406
0
4.62342473×10¹⁵
1.04810413×10²⁹
?
≥1
?
≥1
?
?
161
0
1.32060707×10¹⁹
0
10824
0
0
1.71766739×10²⁹
0
≥1
0
?
0
?
162
0
1.62933648×10¹⁹
3.13609708×10¹⁰
979126
0
8.62077000×10¹⁵
2.82017774×10²⁹
?
≥1
?
≥1
?
?
163
0
2.23153676×10¹⁹
0
16338
0
0
4.62264217×10²⁹
0
≥1
0
?
0
?
164
0
3.36409515×10¹⁹
5.42663231×10¹⁰
1595812
0
1.60791823×10¹⁶
7.58852690×10²⁹
?
≥1
?
≥1
?
?
165
0
4.68906946×10¹⁹
0
22280
0
0
1.24385077×10³⁰
0
≥1
0
?
0
?
166
0
5.86268119×10¹⁹
8.00169996×10¹⁰
1738216
0
2.99935388×10¹⁶
2.04183841×10³⁰
?
≥1
?
≥1
?
?
167
0
7.74386554×10¹⁹
0
30950
0
0
3.34738217×10³⁰
0
≥1
0
?
0
?
168
0
1.13861717×10²⁰
1.35920661×10¹¹
2780870
0
5.59590632×10¹⁶
5.49437301×10³⁰
?
≥1
?
≥1
≥1
?
169
0
1.64107418×10²⁰
0
47344
0
0
9.00729092×10³⁰
0
≥1
0
?
0
?
170
0
2.10998475×10²⁰
2.04853983×10¹¹
3275048
0
1.04419607×10¹⁷
1.47836069×10³¹
?
≥1
?
≥1
?
?
171
0
2.72416255×10²⁰
0
66216
0
0
2.42391112×10³¹
0
≥1
0
?
0
?
172
0
3.88019018×10²⁰
3.43640988×10¹¹
4841772
0
1.94863013×10¹⁷
3.97818200×10³¹
?
≥1
?
≥1
?
?
173
0
5.67763974×10²⁰
0
90580
0
0
6.52253057×10³¹
0
≥1
0
?
0
?
174
0
7.55534710×10²⁰
5.26357426×10¹¹
5913698
0
3.63702584×10¹⁷
1.07041078×10³²
?
≥1
?
≥1
?
?
175
0
9.67627004×10²⁰
0
131634
0
0
1.75519936×10³²
0
≥1
0
?
0
?
176
0
1.33605087×10²¹
8.73742105×10¹¹
8832574
0
6.78891634×10¹⁷
2.88042251×10³²
≥1
≥1
?
≥1
?
?
177
0
1.95161158×10²¹
0
188588
0
0
4.72317330×10³²
0
≥1
0
?
0
?
178
0
2.68270306×10²¹
1.35519434×10¹²
10756252
0
1.26732760×10¹⁸
7.75048327×10³²
?
≥1
?
≥1
?
?
179
0
3.45540249×10²¹
0
263512
0
0
1.27097106×10³³
0
≥1
0
?
0
?
180
0
4.65097189×10²¹
2.23329570×10¹²
15857940
0
2.36604471×10¹⁸
2.08560686×10³³
?
≥1
?
≥1
≥1
?
181
0
6.70187899×10²¹
0
370634
0
0
3.42017029×10³³
0
≥1
0
?
0
?
182
0
9.43762285×10²¹
3.49660746×10¹²
19866066
0
4.41757405×10¹⁸
5.61193956×10³³
?
≥1
?
≥1
?
?
183
0
1.23527382×10²²
0
531350
0
0
9.20332449×10³³
0
≥1
0
?
0
?
184
0
1.63455051×10²²
5.72885702×10¹²
28531702
0
8.24852703×10¹⁸
1.51012600×10³⁴
?
≥1
?
≥1
?
?
185
0
2.30962227×10²²
0
752974
0
0
2.47661371×10³⁴
0
≥1
0
?
0
?
186
0
3.29419563×10²²
9.03248686×10¹²
36626622
0
1.54025886×10¹⁹
4.06350929×10³⁴
?
≥1
?
≥1
?
?
187
0
4.40487573×10²²
0
1041898
0
0
6.66428674×10³⁴
0
≥1
0
?
0
?
188
0
5.78518694×10²²
1.47383634×10¹³
51884754
0
2.87631118×10¹⁹
1.09344795×10³⁵
?
≥1
?
≥1
?
?
189
0
8.00943588×10²²
0
1478840
0
0
1.79335506×10³⁵
0
≥1
0
?
0
?
190
0
1.14425798×10²³
2.33620070×10¹³
67483262
0
5.37156341×10¹⁹
2.94233617×10³⁵
?
≥1
?
≥1
?
?
191
0
1.56300589×10²³
0
2092140
0
0
4.82572185×10³⁵
0
≥1
0
?
0
?
192
0
2.05618038×10²³
3.79972535×10¹³
94933626
0
1.00318857×10²⁰
7.91747704×10³⁵
?
≥1
?
≥1
≥1
?
193
0
2.79729067×10²³
0
2922366
0
0
1.29859141×10³⁶
0
≥1
0
?
0
?
194
0
3.96914225×10²³
6.04644336×10¹³
124708072
0
1.87362671×10²⁰
2.13051816×10³⁶
?
≥1
?
≥1
?
?
195
0
5.51394953×10²³
0
4099820
0
0
3.49438153×10³⁶
0
≥1
0
?
0
?
196
0
7.31813074×10²³
9.81187560×10¹³
174113188
0
3.49945117×10²⁰
5.73295757×10³⁶
?
≥1
?
≥1
?
?
197
0
9.83299276×10²³
0
5777396
0
0
9.40324320×10³⁶
0
≥1
0
?
0
?
198
0
1.37906554×10²⁴
1.56605945×10¹⁴
230622932
0
6.53626735×10²⁰
1.54269338×10³⁷
?
≥1
?
≥1
?
?
199
0
1.93511247×10²⁴
0
8126348
0
0
2.53033242×10³⁷
0
≥1
0
?
0
?
200
0
2.60178811×10²⁴
2.53675473×10¹⁴
321946270
0
1.22087972×10²¹
4.15119627×10³⁷
?
≥1
?
≥1
?
?
N>0
x
all
2k
all
x
2k
all
?
?
?
?
?

Reptile tilings' solutions count (including symmetric)

polyomino \ n²
U pentomino
0
0
0
0
0
0
0
0
0
C hexomino
0
0
0
0
0
0
0
0
?

Smallest common multiples

Smallest known common multiple (area 60):

Common multiples' solutions count (excluding symmetric)

area
30
60
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 $A(w; h)$

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

Idea of proof: U pentominoes should come in pairs to be able to fit cavity in U pentomino (possibly with several C hexominoes inbetween).

Attributions

  1. Smallest known common multiple is found by Giovanni Resta (http://www.iread.it/Poly/P65.html)

See Also

U pentomino and B hexominoU pentomino and D hexomino