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.


C hexomino and I hexomino

Prime rectangles: ≥ 13.

Smallest rectangle tilings

Smallest rectangle (10x12):

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
1-9
10
11
12
13
14
15
16
17
18
19
20
N>0
1-9
0
10
0
0
11
0
0
0
12
0
2
4
12
13
0
0
0
20
0
14
0
0
0
28
0
0
15
0
0
0
36
0
?
0
16
0
0
0
102
0
0
?
0
17
0
0
0
212
0
0
0
0
0
18
0
74
160
628
1248
?
?
?
?
?
19
0
0
0
1248
0
0
0
0
0
?
0
20
0
0
0
2104
0
0
?
0
0
≥1
0
0
21
0
0
0
3228
0
?
0
?
0
≥1
0
?
?
22
0
0
0
5904
0
0
?
0
0
≥1
0
0
?
23
0
0
0
10786
0
0
0
0
0
≥1
0
0
?
24
0
1630
3862
25488
65280
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
25
0
0
0
52282
0
0
0
0
0
≥1
0
0
?
26
0
0
0
96604
0
0
?
0
0
≥1
0
0
?
27
0
0
0
165050
0
?
0
?
0
?
0
?
?
28
0
0
0
289692
0
0
?
0
0
?
0
0
?
29
0
0
0
508286
0
0
0
0
0
?
0
0
?
30
0
28564
74612
1015494
3344012
?
?
?
?
≥1
?
≥1
?
31
0
0
0
1996002
0
0
0
0
0
≥1
0
0
?
32
0
0
0
3765080
0
0
?
0
0
≥1
0
0
?
33
0
0
0
6771906
0
?
0
?
0
≥1
0
?
?
34
0
0
0
12075112
0
0
?
0
0
≥1
0
0
?
35
0
0
0
21298536
0
0
0
0
0
≥1
0
0
?
36
0
441834
1277330
39584046
164486188
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
37
0
0
0
74384218
0
0
0
0
0
≥1
0
0
?
38
0
0
0
138696656
0
0
?
0
0
≥1
0
0
?
39
0
0
0
253206454
0
?
0
?
0
≥1
0
?
?
40
0
0
0
457243500
0
0
?
0
0
≥1
0
0
?
41
0
0
0
816067682
0
0
0
0
0
≥1
0
0
?
42
0
6328898
20325412
1.48146153×10¹⁰
7.75665994×10¹⁰
?
?
?
?
≥1
?
≥1
?
43
0
0
0
2.71345689×10¹⁰
0
0
0
0
0
≥1
0
0
?
44
0
0
0
4.97986345×10¹⁰
0
0
?
0
0
≥1
0
0
?
45
0
0
0
9.07769600×10¹⁰
0
?
0
?
0
≥1
0
?
?
46
0
0
0
1.64474463×10¹¹
0
0
?
0
0
≥1
0
0
?
47
0
0
0
2.95603599×10¹¹
0
0
0
0
0
≥1
0
0
?
48
0
86163644
308581586
5.33275944×10¹¹
3.54527872×10¹²
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
49
0
0
0
9.65662406×10¹¹
0
0
0
0
0
≥1
0
0
?
50
0
0
0
1.75303098×10¹²
0
0
?
0
0
≥1
0
0
?
51
0
0
0
3.17809526×10¹²
0
?
0
?
0
≥1
0
?
?
52
0
0
0
5.74812439×10¹²
0
0
?
0
0
≥1
0
0
?
53
0
0
0
1.03511723×10¹³
0
0
0
0
0
≥1
0
0
?
54
0
1.13255113×10¹⁰
4.54021371×10¹⁰
1.86381792×10¹³
1.58552704×10¹⁴
?
?
?
?
≥1
?
≥1
?
55
0
0
0
3.35883290×10¹³
0
0
0
0
0
≥1
0
0
?
56
0
0
0
6.06046236×10¹³
0
0
?
0
0
≥1
0
0
?
57
0
0
0
1.09348097×10¹⁴
0
?
0
?
0
≥1
0
?
?
58
0
0
0
1.97161948×10¹⁴
0
0
?
0
0
≥1
0
0
?
59
0
0
0
3.54780268×10¹⁴
0
0
0
0
0
≥1
0
0
?
60
0
1.45184393×10¹¹
6.53768130×10¹¹
6.37929020×10¹⁴
6.97923904×10¹⁵
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
61
0
0
0
1.14683849×10¹⁵
0
0
0
0
0
≥1
0
0
?
62
0
0
0
2.06233105×10¹⁵
0
0
?
0
0
≥1
0
0
?
63
0
0
0
3.70876296×10¹⁵
0
?
0
?
0
≥1
0
?
?
64
0
0
0
6.66839056×10¹⁵
0
0
?
0
0
≥1
0
0
?
65
0
0
0
1.19793938×10¹⁶
0
0
0
0
0
≥1
0
0
?
66
0
1.82755724×10¹²
9.27233034×10¹²
2.15077923×10¹⁶
3.03468341×10¹⁷
?
?
?
?
≥1
?
≥1
?
67
0
0
0
3.85999923×10¹⁶
0
0
0
0
0
≥1
0
0
?
68
0
0
0
6.92675652×10¹⁶
0
0
?
0
0
≥1
0
0
?
69
0
0
0
1.24287908×10¹⁷
0
?
0
?
0
≥1
0
?
?
70
0
0
0
2.22986108×10¹⁷
0
0
?
0
0
≥1
0
0
?
71
0
0
0
3.99898850×10¹⁷
0
0
0
0
0
≥1
0
0
?
72
0
2.26971238×10¹³
1.30084805×10¹⁴
7.16903911×10¹⁷
1.30654596×10¹⁹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
73
0
0
0
1.28477221×10¹⁸
0
0
0
0
0
≥1
0
0
?
74
0
0
0
2.30195682×10¹⁸
0
0
?
0
0
≥1
0
0
?
75
0
0
0
4.12375074×10¹⁸
0
?
0
?
0
≥1
0
?
?
76
0
0
0
7.38630643×10¹⁸
0
0
?
0
0
≥1
0
0
?
77
0
0
0
1.32268867×10¹⁹
0
0
0
0
0
≥1
0
0
?
78
0
2.79052082×10¹⁴
1.81052321×10¹⁵
2.36799506×10¹⁹
5.57943940×10²⁰
?
?
?
?
≥1
?
≥1
?
79
0
0
0
4.23833210×10¹⁹
0
0
0
0
0
≥1
0
0
?
80
0
0
0
7.58434333×10¹⁹
0
0
?
0
0
≥1
0
0
?
81
0
0
0
1.35694013×10²⁰
0
?
0
?
0
≥1
0
?
?
82
0
0
0
2.42737456×10²⁰
0
0
?
0
0
≥1
0
0
?
83
0
0
0
4.34144353×10²⁰
0
0
0
0
0
≥1
0
0
?
84
0
3.40467295×10¹⁵
2.50499085×10¹⁶
7.76339252×10²⁰
2.36635626×10²²
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
85
0
0
0
1.38798940×10²¹
0
0
0
0
0
≥1
0
0
?
86
0
0
0
2.48109513×10²¹
0
0
?
0
0
≥1
0
0
?
87
0
0
0
4.43432987×10²¹
0
?
0
?
0
≥1
0
?
?
88
0
0
0
7.92406376×10²¹
0
0
?
0
0
≥1
0
0
?
89
0
0
0
1.41579850×10²²
0
0
0
0
0
≥1
0
0
?
90
0
4.12971468×10¹⁶
3.45035153×10¹⁷
2.52923573×10²²
9.97786646×10²³
?
?
?
?
≥1
?
≥1
?
91
0
0
0
4.51763011×10²²
0
0
0
0
0
≥1
0
0
?
92
0
0
0
8.06802952×10²²
0
0
?
0
0
≥1
0
0
?
93
0
0
0
1.44066050×10²³
0
?
0
?
0
≥1
0
?
?
94
0
0
0
2.57215614×10²³
0
0
?
0
0
≥1
0
0
?
95
0
0
0
4.59171787×10²³
0
0
0
0
0
≥1
0
0
?
96
0
4.98652232×10¹⁷
4.73624695×10¹⁸
8.19590110×10²³
4.18616498×10²⁵
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
97
0
0
0
1.46272359×10²⁴
0
0
0
0
0
≥1
0
0
?
98
0
0
0
2.61019568×10²⁴
0
0
?
0
0
≥1
0
0
?
99
0
0
0
4.65725663×10²⁴
0
?
0
?
0
≥1
0
?
?
100
0
0
0
8.30875104×10²⁴
0
0
?
0
0
≥1
0
0
?
N>0
x
6k
6k
all
6k
?
?
?
?
?
?
?

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 $A(w; h)$

$G(N(10); x) = \frac{2x^{12} - 46x^{18} + 450x^{24} - 2310x^{30} + 5022x^{36} + 13406x^{42} - 147762x^{48} + 568230x^{54} - 1128668x^{60} + 79620x^{66} + 7862212x^{72} - 31891404x^{78} + 80603384x^{84} - 152394036x^{90} + 227687984x^{96} - 273438500x^{102} + 262264654x^{108} - 193268142x^{114} + 95720870x^{120} - 10357734x^{126} - 36343486x^{132} + 43967390x^{138} - 29934334x^{144} + 12875190x^{150} - 2242914x^{156} - 1523738x^{162} + 1660768x^{168} - 892188x^{174} + 329722x^{180} - 88858x^{186} + 17384x^{192} - 2360x^{198} + 200x^{204} - 8x^{210}}{1 - 60x^6 + 1630x^{12} - 26847x^{18} + 303403x^{24} - 2527992x^{30} + 16307640x^{36} - 84446149x^{42} + 361024584x^{48} - 1303066286x^{54} + 4042715944x^{60} - 10938268282x^{66} + 26113184245x^{72} - 55523659222x^{78} + 105940134904x^{84} - 182472216969x^{90} + 285052784353x^{96} - 405345974688x^{102} + 526128321757x^{108} - 624575848664x^{114} + 679016257585x^{120} - 676538333124x^{126} + 617874949283x^{132} - 517081356960x^{138} + 396188686741x^{144} - 277551734780x^{150} + 177447292275x^{156} - 103280946006x^{162} + 54560026056x^{168} - 26062629610x^{174} + 11207405848x^{180} - 4315207904x^{186} + 1478113060x^{192} - 446929358x^{198} + 118155647x^{204} - 26989413x^{210} + 5246424x^{216} - 850686x^{222} + 111930x^{228} - 11480x^{234} + 861x^{240} - 42x^{246} + x^{252}} \tag{1}$

See Also

Z pentomino and Y2 hexominoI hexomino and T1 hexomino