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.


I triomino and I tetromino

Prime rectangles: 16.

Smallest rectangle tilings

Smallest rectangle (1x7):

Smallest square (4x4):

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
0
3
0
0
0
4
0
0
0
12
5
0
0
2
23
16
6
0
0
3
40
101
348
7
2
4
12
172
778
3256
23880
8
0
0
23
452
1820
12241
143972
≥1
9
0
0
39
981
4375
38404
729116
≥1
≥1
10
3
9
84
2694
23566
217113
4231279
≥1
≥1
≥1
11
3
9
177
7281
84392
1115324
26806249
≥1
≥1
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≥1
12
0
2
318
≥1
≥1
≥1
≥1
≥1
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≥1
≥1
13
4
16
575
≥1
≥1
≥1
≥1
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≥1
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≥1
14
6
36
1176
≥1
≥1
≥1
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≥1
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15
4
24
2241
≥1
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≥1
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16
5
35
4040
≥1
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17
10
100
7710
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18
10
120
14916
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19
11
121
27681
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20
15
255
51342
≥1
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≥1
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≥1
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21
20
440
98195
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22
22
484
185559
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23
27
729
345033
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24
35
1367
650210
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25
43
1849
1233258
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26
49
2401
2314041
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27
63
4095
4338090
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≥1
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28
79
6399
8199096
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?
29
92
8464
15465413
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≥1
≥1
≥1
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?
30
112
12768
29038038
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?
31
144
20736
54692835
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?
32
171
29583
103247129
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?
33
204
42024
194387532
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?
34
257
66049
365795196
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?
35
316
99856
689836875
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36
375
142127
1.30060726×10¹⁰
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37
462
213444
2.44904561×10¹⁰
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38
573
328329
4.61478418×10¹⁰
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?
39
692
480248
8.70196256×10¹⁰
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?
40
838
703920
1.63986439×10¹¹
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?
41
1035
1071225
3.08963709×10¹¹
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?
42
1265
1602755
5.82440248×10¹¹
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?
43
1532
2347024
1.09798773×10¹²
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44
1873
3511875
2.06916688×10¹²
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45
2300
5294600
3.89993945×10¹²
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?
46
2798
7828804
7.35213189×10¹²
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≥1
≥1
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?
47
3406
11600836
1.38581653×10¹³
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≥1
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48
4173
17430623
2.61195575×10¹³
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?
49
5099
25999801
4.92366165×10¹³
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50
6204
38489616
9.28151827×10¹³
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51
7580
57471560
1.74948713×10¹⁴
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52
9273
86007075
3.29772943×10¹⁴
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53
11303
127757809
6.21650247×10¹⁴
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54
13784
190026224
1.17182681×10¹⁵
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55
16855
284091025
2.20887039×10¹⁵
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56
20576
423412928
4.16382617×10¹⁵
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?
57
25087
629407743
7.84908716×10¹⁵
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?
58
30640
938809600
1.47957031×10¹⁶
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?
59
37432
1.40115462×10¹⁰
2.78903635×10¹⁶
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60
45663
2.08529222×10¹⁰
5.25751183×10¹⁶
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61
55728
3.10560998×10¹⁰
9.91068162×10¹⁶
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62
68072
4.63379718×10¹⁰
1.86820085×10¹⁷
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63
83096
6.90511140×10¹⁰
3.52165985×10¹⁷
≥1
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64
101392
1.02805404×10¹¹
6.63854411×10¹⁷
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65
123800
1.53264400×10¹¹
1.25139879×10¹⁸
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66
151168
2.28520665×10¹¹
2.35895066×10¹⁸
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67
184490
3.40365601×10¹¹
4.44676362×10¹⁸
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68
225192
5.07118872×10¹¹
8.38240643×10¹⁸
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69
274968
7.56079509×10¹¹
1.58012861×10¹⁹
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70
335659
1.12666964×10¹²
2.97863348×10¹⁹
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71
409683
1.67840160×10¹²
5.61490272×10¹⁹
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72
500160
2.50162026×10¹²
1.05844129×10²⁰
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73
610628
3.72866554×10¹²
1.99522234×10²⁰
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74
745342
5.55534696×10¹²
3.76111294×10²⁰
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75
909844
8.27817924×10¹²
7.08992069×10²⁰
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76
1110789
1.23385442×10¹³
1.33649122×10²¹
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77
1355970
1.83865464×10¹³
2.51936477×10²¹
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?
78
1655186
2.73964400×10¹³
4.74915307×10²¹
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?
79
2020635
4.08296580×10¹³
8.95243405×10²¹
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80
2466759
6.08490489×10¹³
1.68758660×10²²
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?
81
3011156
9.06706647×10¹³
3.18120131×10²²
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?
82
3675822
1.35116673×10¹⁴
5.99675401×10²²
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?
83
4487395
2.01367138×10¹⁴
1.13042366×10²³
≥1
≥1
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?
84
5477915
3.00075746×10¹⁴
2.13091583×10²³
≥1
≥1
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?
85
6686979
4.47156881×10¹⁴
4.01690352×10²³
≥1
≥1
≥1
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≥1
≥1
≥1
≥1
≥1
≥1
≥1
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?
86
8163217
6.66381117×10¹⁴
7.57210229×10²³
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
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?
87
9965311
9.93074432×10¹⁴
1.42738632×10²⁴
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
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≥1
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?
88
12164895
1.47984694×10¹⁵
2.69070836×10²⁴
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
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≥1
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?
89
14850196
2.20528321×10¹⁵
5.07214581×10²⁴
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
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≥1
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≥1
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?
90
18128528
3.28643563×10¹⁵
9.56129704×10²⁴
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
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?
91
22130208
4.89746106×10¹⁵
1.80236150×10²⁵
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
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?
92
27015091
7.29815195×10¹⁵
3.39755900×10²⁵
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
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≥1
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≥1
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≥1
?
93
32978724
1.08759630×10¹⁶
6.40460141×10²⁵
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
94
40258737
1.62076590×10¹⁶
1.20730556×10²⁶
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
95
49145300
2.41526051×10¹⁶
2.27584303×10²⁶
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
96
59993815
3.59925807×10¹⁶
4.29009995×10²⁶
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
97
73237462
5.36372584×10¹⁶
8.08709437×10²⁶
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
98
89404037
7.99308183×10¹⁶
1.52446554×10²⁷
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
99
109139116
1.19113468×10¹⁷
2.87370855×10²⁷
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
100
133231278
1.77505737×10¹⁷
5.41711214×10²⁷
≥1
≥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
?
?
?
?
?
?
?
?
?
?
?
?
?

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 $A(w; h)$

$G(N(1); x) = \frac{2x^7 + 2x^8 + 2x^9 + x^{10}}{1 + x + x^2 - x^3 - 3x^4 - 3x^5 - 2x^6 + x^7 + 2x^8 + 2x^9 + x^{10}} \tag{1}$

$G(N(2); x) = \frac{4x^7 + 4x^8 - 3x^{10} - 6x^{11} - 5x^{12} - x^{13} - x^{14} + x^{16}}{1 + x - 3x^3 - 6x^4 - 4x^5 + 2x^6 + 7x^7 + 9x^8 + 5x^9 - x^{10} - 4x^{11} - 5x^{12} - 2x^{13} - x^{14} + x^{16}} \tag{2}$

$G(N(3); x) = \frac{2x^5 - 3x^6 + 9x^7 - 14x^8 - x^9 + 4x^{10} - 17x^{11} + 10x^{12} + x^{13} - 31x^{14} + 53x^{15} - 2x^{16} - 18x^{17} + 48x^{18} - 49x^{19} + 22x^{20} + x^{21} - 10x^{22} + 15x^{23} - 3x^{24} + x^{25} - x^{26} - x^{27}}{1 - 3x + 3x^2 - 5x^3 + 4x^4 + 8x^5 - 8x^6 + x^7 + 11x^8 - 30x^9 + 36x^{10} - 14x^{11} - 46x^{12} + 67x^{13} - 80x^{14} + 49x^{15} + 23x^{16} - 46x^{17} + 61x^{18} - 46x^{19} + 9x^{20} + 3x^{21} - 11x^{22} + 16x^{23} - 2x^{24} + x^{25} - x^{26} - x^{27}} \tag{3}$

See Also

Dominoes and T1 hexominoI triomino and L tetromino