POLYOMINO TILINGS

Polyomino Tilings

Select polyominoes for a set (currently 1 or 2), for which tilings should be shown.

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You may also see list of all polyomino sets for which data is available here.


P2 9-omino

Area: 9.

Size: 3x5.

Is rectangular: no.

Is convex: yes.

Holes: 0.

Order: 4.

Square order: 4.

Odd order: ≤ 87.

Prime rectangles: ≥ 26.

Smallest rectangle tilings

Smallest rectangle and smallest square (6x6):

Smallest known odd rectangle (27x29):

Rectangle tilings' solutions count (including symmetric)

Blue number (P) - strongly prime rectangle (which cannot be divided into two or more number of rectangles tileable by this set).

Green number (W) - weakly prime rectangle (which cannot be divided into two rectangles tileable by this set, but which can be divided into three or more rectangles).

Red number (C) - 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 \ h1-567-111213-141516171819-2021222324252627N>0
1-50
6022P
7-11000
12044C01616C
13-1400000
150001616P00
160000000
1700000000
18088C06464C00000512512C
19-2000000000000
21000384384C0000000??
2200000000000????
2300000000000??????
2401616C0256256C0256256C0040964096C0≥1≥1C????≥1≥1C
2500000000000??????????
2600000000000????????????
2700069126912C00000000??≥1≥1P??≥1≥1C??????
2800000000000???????????????
2900000000000????????????≥1≥1P?
3003232C012801280C00004915249152C0??????≥1≥1C???????
3100000000000???????????????
3200000000000???????????????
33000114688114688C0000000??????≥1≥1C???????
3400000000000???????????????
3500000000000???????????????
3606464C01536015360C040964096C000126824960126824960C0≥1≥1C????≥1≥1C???????
3700000000000???????????????
3800000000000???????????????
3900018554881855488C0000000???????????????
4000000000000???????????????
4100000000000???????????????
420128128C0335872335872C00004.44053995×10¹¹44405399552C0???????????????
4300000000000???????????????
4400000000000???????????????
450002979020829790208C00000000???????????????
4600000000000???????????????
4700000000000???????????????
480256256C075694087569408C06553665536C005.03430014×10¹³5034300145664C0≥1≥1C?????????????
4900000000000???????????????
5000000000000???????????????
51000477298688477298688C0000000???????????????
52000000004.38957563×10¹²438957563904P0???????????????
5300000000000???????????????
540512512C0159580160159580160C000003.88849124×10¹⁵388849124507648C0???????????????
5500000000000???????????????
5600000000000???????????????
570007.64477440×10¹⁰7644774400C0000000???????????????
58000000002.49799407×10¹⁵249799407058944C0???????????????
5900000000000???????????????
60010241024C03.18380441×10¹⁰3183804416C010485761048576C002.45333429×10¹⁷24533342932172800C0≥1≥1C?????????????
6100000000000???????????????
6200000000000???????????????
630001.22469482×10¹²122469482496C00000000???????????????
64000000004.10920702×10¹⁷41092070296780800C0???????????????
6500000000000???????????????
66020482048C06.10997370×10¹¹61099737088C00001.36456434×10¹⁹1364564349909499904C0???????????????
6700000000000???????????????
68000000001.28515079×10¹⁶1285150792089600P0???????????????
690001.96267737×10¹³1962677370880C00001.14920003×10¹⁶1149200039411712P0≥1≥1P?????????????
70000000004.26394239×10¹⁹4263942395384561664C0???????????????
7100000000000???????????????
72040964096C01.14054764×10¹³1140547649536C01677721616777216C000≥1.84467440×10²⁰≥18446744073709551615C0≥1≥1C≥1≥1P???????????
7300000000000???????????????
74000000009.12480741×10¹⁸912480741762269184C0???????????????
750003.14664414×10¹⁴31466441408512C00002.92106237×10¹⁸292106237468737536C0≥1≥1P?????????????
7600000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
7700000000000???????????????
78081928192C02.08600815×10¹⁴20860081537024C0000≥1.84467440×10²⁰≥18446744073709551615C0≥1≥1P?????????????
7900000000000???????????????
8000000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
810005.04695072×10¹⁵504695072227328C00000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
8200000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
8300000000000???????????????
8401638416384C03.75609419×10¹⁵375609419104256C0268435456268435456C00≥1.84467440×10²⁰≥18446744073709551615C0≥1≥1C?????????????
85000000005.58741854×10¹⁹5587418544561192960P0???????????????
8600000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
870008.09829795×10¹⁶8098297956794368C0000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
8800000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
8900000000000???????????????
9003276832768C06.68060622×10¹⁶6680606226251776C00000≥1.84467440×10²⁰≥18446744073709551615C0??≥1≥1P???????????
9100000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
9200000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
930001.29999235×10¹⁸129999235817930752C0000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
9400000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
95000000002.35356168×10¹⁹2353561680715186176P0???????????????
9606553665536C01.17648242×10¹⁸117648242858000384C04.29496729×10¹⁰4294967296C00≥1.84467440×10²⁰≥18446744073709551615C0≥1≥1C?????????????
9700000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
9800000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
990002.08770339×10¹⁹2087703397829443584C00000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
10000000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
10100000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
1020131072131072C02.05497882×10¹⁹2054978825988603904C0000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
10300000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
10400000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
105000≥1.84467440×10²⁰≥18446744073709551615C0000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
10600000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
10700000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
1080262144262144C0≥1.84467440×10²⁰≥18446744073709551615C06.87194767×10¹¹68719476736C01.00306130×10¹⁴10030613004288P≥1.84467440×10²⁰≥18446744073709551615C0≥1≥1C?????????????
10900000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
11000000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
111000≥1.84467440×10²⁰≥18446744073709551615C0000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
11200000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
11300000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
1140524288524288C0≥1.84467440×10²⁰≥18446744073709551615C0000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
11500000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
11600000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
117000≥1.84467440×10²⁰≥18446744073709551615C00000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
11800000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
11900000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
120010485761048576C0≥1.84467440×10²⁰≥18446744073709551615C01.09951162×10¹³1099511627776C00≥1.84467440×10²⁰≥18446744073709551615C0???????????????
12100000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
12200000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
123000≥1.84467440×10²⁰≥18446744073709551615C0000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
12400000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
12500000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
126020971522097152C0≥1.84467440×10²⁰≥18446744073709551615C00000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
12700000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
12800000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
129000≥1.84467440×10²⁰≥18446744073709551615C0000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
13000000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
13100000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
132041943044194304C0≥1.84467440×10²⁰≥18446744073709551615C01.75921860×10¹⁴17592186044416C00≥1.84467440×10²⁰≥18446744073709551615C0???????????????
13300000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
13400000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
135000≥1.84467440×10²⁰≥18446744073709551615C00000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
13600000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
13700000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
138083886088388608C0≥1.84467440×10²⁰≥18446744073709551615C0000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
13900000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
14000000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
141000≥1.84467440×10²⁰≥18446744073709551615C0000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
14200000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
14300000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
14401677721616777216C0≥1.84467440×10²⁰≥18446744073709551615C02.81474976×10¹⁵281474976710656C000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
14500000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
14600000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
147000≥1.84467440×10²⁰≥18446744073709551615C0000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
14800000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
14900000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
15003355443233554432C0≥1.84467440×10²⁰≥18446744073709551615C0000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
15100000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
15200000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
153000≥1.84467440×10²⁰≥18446744073709551615C00000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
15400000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
15500000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
15606710886467108864C0≥1.84467440×10²⁰≥18446744073709551615C04.50359962×10¹⁶4503599627370496C00≥1.84467440×10²⁰≥18446744073709551615C0???????????????
15700000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
15800000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
159000≥1.84467440×10²⁰≥18446744073709551615C0000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
16000000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
16100000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
1620134217728134217728C0≥1.84467440×10²⁰≥18446744073709551615C00000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
16300000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
16400000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
165000≥1.84467440×10²⁰≥18446744073709551615C0000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
16600000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
16700000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
1680268435456268435456C0≥1.84467440×10²⁰≥18446744073709551615C07.20575940×10¹⁷72057594037927936C00≥1.84467440×10²⁰≥18446744073709551615C0???????????????
16900000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
17000000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
171000≥1.84467440×10²⁰≥18446744073709551615C00000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
17200000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
17300000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
1740536870912536870912C0≥1.84467440×10²⁰≥18446744073709551615C0000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
17500000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
17600000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
177000≥1.84467440×10²⁰≥18446744073709551615C0000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
17800000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
17900000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
18001.07374182×10¹⁰1073741824C0≥1.84467440×10²⁰≥18446744073709551615C01.15292150×10¹⁹1152921504606846976C0≥1.84467440×10²⁰≥18446744073709551615P≥1.84467440×10²⁰≥18446744073709551615C0???????????????
18100000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
18200000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
183000≥1.84467440×10²⁰≥18446744073709551615C0000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
18400000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
18500000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
18602.14748364×10¹⁰2147483648C0≥1.84467440×10²⁰≥18446744073709551615C0000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
18700000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
18800000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
189000≥1.84467440×10²⁰≥18446744073709551615C00000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
19000000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
19100000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
19204.29496729×10¹⁰4294967296C0≥1.84467440×10²⁰≥18446744073709551615C0≥1.84467440×10²⁰≥18446744073709551615C00≥1.84467440×10²⁰≥18446744073709551615C0???????????????
19300000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
19400000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
195000≥1.84467440×10²⁰≥18446744073709551615C0000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
19600000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
19700000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
19808.58993459×10¹⁰8589934592C0≥1.84467440×10²⁰≥18446744073709551615C00000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
19900000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
20000000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
201000≥1.84467440×10²⁰≥18446744073709551615C0000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
20200000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
20300000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
20401.71798691×10¹¹17179869184C0≥1.84467440×10²⁰≥18446744073709551615C0≥1.84467440×10²⁰≥18446744073709551615C00≥1.84467440×10²⁰≥18446744073709551615C0???????????????
20500000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
20600000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
207000≥1.84467440×10²⁰≥18446744073709551615C00000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
20800000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
20900000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
21003.43597383×10¹¹34359738368C0≥1.84467440×10²⁰≥18446744073709551615C0000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
21100000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
21200000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
213000≥1.84467440×10²⁰≥18446744073709551615C0000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
21400000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
21500000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
21606.87194767×10¹¹68719476736C0≥1.84467440×10²⁰≥18446744073709551615C0≥1.84467440×10²⁰≥18446744073709551615C0≥1.84467440×10²⁰≥18446744073709551615C≥1.84467440×10²⁰≥18446744073709551615C0???????????????
21700000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
21800000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
219000≥1.84467440×10²⁰≥18446744073709551615C0000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
22000000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
22100000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
22201.37438953×10¹²137438953472C0≥1.84467440×10²⁰≥18446744073709551615C0000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
22300000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
22400000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
225000≥1.84467440×10²⁰≥18446744073709551615C00000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
22600000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
22700000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
22802.74877906×10¹²274877906944C0≥1.84467440×10²⁰≥18446744073709551615C0≥1.84467440×10²⁰≥18446744073709551615C00≥1.84467440×10²⁰≥18446744073709551615C0???????????????
22900000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
23000000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
231000≥1.84467440×10²⁰≥18446744073709551615C0000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
23200000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
23300000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
23405.49755813×10¹²549755813888C0≥1.84467440×10²⁰≥18446744073709551615C000≥1.84467440×10²⁰≥18446744073709551615P≥1.84467440×10²⁰≥18446744073709551615C0???????????????
23500000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
23600000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
237000≥1.84467440×10²⁰≥18446744073709551615C0000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
23800000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
23900000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
24001.09951162×10¹³1099511627776C0≥1.84467440×10²⁰≥18446744073709551615C0≥1.84467440×10²⁰≥18446744073709551615C00≥1.84467440×10²⁰≥18446744073709551615C0???????????????
24100000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
24200000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
243000≥1.84467440×10²⁰≥18446744073709551615C000≥1.84467440×10²⁰≥18446744073709551615P≥1.84467440×10²⁰≥18446744073709551615C0???????????????
24400000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
24500000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
24602.19902325×10¹³2199023255552C0≥1.84467440×10²⁰≥18446744073709551615C0000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
24700000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
24800000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
249000≥1.84467440×10²⁰≥18446744073709551615C0000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
25000000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
25100000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
25204.39804651×10¹³4398046511104C0≥1.84467440×10²⁰≥18446744073709551615C0≥1.84467440×10²⁰≥18446744073709551615C0≥1.84467440×10²⁰≥18446744073709551615P≥1.84467440×10²⁰≥18446744073709551615C0???????????????
25300000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
25400000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
255000≥1.84467440×10²⁰≥18446744073709551615C0000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
25600000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
25700000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
25808.79609302×10¹³8796093022208C0≥1.84467440×10²⁰≥18446744073709551615C0000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
25900000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
26000000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
261000≥1.84467440×10²⁰≥18446744073709551615C000≥1.84467440×10²⁰≥18446744073709551615P≥1.84467440×10²⁰≥18446744073709551615C0???????????????
26200000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
26300000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
26401.75921860×10¹⁴17592186044416C0≥1.84467440×10²⁰≥18446744073709551615C0≥1.84467440×10²⁰≥18446744073709551615C00≥1.84467440×10²⁰≥18446744073709551615C0???????????????
26500000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
26600000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
267000≥1.84467440×10²⁰≥18446744073709551615C0000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
26800000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
26900000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
27003.51843720×10¹⁴35184372088832C0≥1.84467440×10²⁰≥18446744073709551615C000≥1.84467440×10²⁰≥18446744073709551615P≥1.84467440×10²⁰≥18446744073709551615C0???????????????
27100000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
27200000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
273000≥1.84467440×10²⁰≥18446744073709551615C0000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
27400000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
27500000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
27607.03687441×10¹⁴70368744177664C0≥1.84467440×10²⁰≥18446744073709551615C0≥1.84467440×10²⁰≥18446744073709551615C00≥1.84467440×10²⁰≥18446744073709551615C0???????????????
27700000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
27800000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
279000≥1.84467440×10²⁰≥18446744073709551615C000≥1.84467440×10²⁰≥18446744073709551615P≥1.84467440×10²⁰≥18446744073709551615C0???????????????
28000000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
28100000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
28201.40737488×10¹⁵140737488355328C0≥1.84467440×10²⁰≥18446744073709551615C0000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
28300000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
28400000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
285000≥1.84467440×10²⁰≥18446744073709551615C0000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
28600000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
28700000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
28802.81474976×10¹⁵281474976710656C0≥1.84467440×10²⁰≥18446744073709551615C0≥1.84467440×10²⁰≥18446744073709551615C0≥1.84467440×10²⁰≥18446744073709551615C≥1.84467440×10²⁰≥18446744073709551615C0???????????????
28900000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
29000000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
291000≥1.84467440×10²⁰≥18446744073709551615C0000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
29200000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
29300000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
29405.62949953×10¹⁵562949953421312C0≥1.84467440×10²⁰≥18446744073709551615C0000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
29500000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
29600000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
297000≥1.84467440×10²⁰≥18446744073709551615C000≥1.84467440×10²⁰≥18446744073709551615P≥1.84467440×10²⁰≥18446744073709551615C0???????????????
29800000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
29900000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
30001.12589990×10¹⁶1125899906842624C0≥1.84467440×10²⁰≥18446744073709551615C0≥1.84467440×10²⁰≥18446744073709551615C00≥1.84467440×10²⁰≥18446744073709551615C0???????????????
30100000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
30200000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
303000≥1.84467440×10²⁰≥18446744073709551615C0000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
30400000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
30500000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
30602.25179981×10¹⁶2251799813685248C0≥1.84467440×10²⁰≥18446744073709551615C000≥1.84467440×10²⁰≥18446744073709551615P≥1.84467440×10²⁰≥18446744073709551615C0???????????????
30700000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
30800000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
309000≥1.84467440×10²⁰≥18446744073709551615C0000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
31000000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
31100000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
31204.50359962×10¹⁶4503599627370496C0≥1.84467440×10²⁰≥18446744073709551615C0≥1.84467440×10²⁰≥18446744073709551615C00≥1.84467440×10²⁰≥18446744073709551615C0???????????????
31300000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
31400000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
315000≥1.84467440×10²⁰≥18446744073709551615C000≥1.84467440×10²⁰≥18446744073709551615P≥1.84467440×10²⁰≥18446744073709551615C0???????????????
31600000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
31700000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
31809.00719925×10¹⁶9007199254740992C0≥1.84467440×10²⁰≥18446744073709551615C0000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
31900000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
32000000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
321000≥1.84467440×10²⁰≥18446744073709551615C0000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
32200000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
32300000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
32401.80143985×10¹⁷18014398509481984C0≥1.84467440×10²⁰≥18446744073709551615C0≥1.84467440×10²⁰≥18446744073709551615C0≥1.84467440×10²⁰≥18446744073709551615C≥1.84467440×10²⁰≥18446744073709551615C0???????????????
32500000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
32600000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
327000≥1.84467440×10²⁰≥18446744073709551615C0000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
32800000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
32900000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
33003.60287970×10¹⁷36028797018963968C0≥1.84467440×10²⁰≥18446744073709551615C0000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
33100000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
33200000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
333000≥1.84467440×10²⁰≥18446744073709551615C000≥1.84467440×10²⁰≥18446744073709551615P≥1.84467440×10²⁰≥18446744073709551615C0???????????????
33400000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
33500000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
33607.20575940×10¹⁷72057594037927936C0≥1.84467440×10²⁰≥18446744073709551615C0≥1.84467440×10²⁰≥18446744073709551615C00≥1.84467440×10²⁰≥18446744073709551615C0???????????????
33700000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
33800000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
339000≥1.84467440×10²⁰≥18446744073709551615C0000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
34000000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
34100000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
34201.44115188×10¹⁸144115188075855872C0≥1.84467440×10²⁰≥18446744073709551615C000≥1.84467440×10²⁰≥18446744073709551615C≥1.84467440×10²⁰≥18446744073709551615C0???????????????
34300000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
34400000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
345000≥1.84467440×10²⁰≥18446744073709551615C0000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
34600000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
34700000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
34802.88230376×10¹⁸288230376151711744C0≥1.84467440×10²⁰≥18446744073709551615C0≥1.84467440×10²⁰≥18446744073709551615C00≥1.84467440×10²⁰≥18446744073709551615C0???????????????
34900000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
35000000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
351000≥1.84467440×10²⁰≥18446744073709551615C000≥1.84467440×10²⁰≥18446744073709551615C≥1.84467440×10²⁰≥18446744073709551615C0???????????????
35200000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
35300000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
35405.76460752×10¹⁸576460752303423488C0≥1.84467440×10²⁰≥18446744073709551615C0000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
35500000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
35600000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
357000≥1.84467440×10²⁰≥18446744073709551615C0000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
35800000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
35900000000≥1.84467440×10²⁰≥18446744073709551615C0???????????????
36001.15292150×10¹⁹1152921504606846976C0≥1.84467440×10²⁰≥18446744073709551615C0≥1.84467440×10²⁰≥18446744073709551615C0≥1.84467440×10²⁰≥18446744073709551615C≥1.84467440×10²⁰≥18446744073709551615C0???????????????
N>0x6kx3kx12kx9kallx???????

Smallest prime reptiles

Smallest prime reptile (9P2x6):

Reptile tilings' solutions count (including symmetric)

polyomino \ n²
P2 9-omino10000512P00

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 $A(w; h)$

$N(1; n) = T(1; n) = 0, \qquad n \geq 1 \tag{1}$

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

P2 enneomino has width and height greater than $2$ thus it does not fit into $1 \times n$ and $2 \times n$ rectangles. Q.E.D.

$N(3; n) = T(3; n) = 0, \qquad n \geq 1 \tag{3}$

If P2 enneomino tiles $3 \times n$ rectangle then all rectangle cells should filled. Top left corner (marked with dot on image) may be covered in only three ways. All of them have untileable holes. Q.E.D.

$N(4; n) = T(4; n) = 0, \qquad n \geq 1 \tag{4}$

If P2 enneomino tiles $4 \times n$ rectangle then all rectangle cells should filled. Top left corner (marked with dot on image) may be covered in only three ways. All of them have untileable holes. Q.E.D.

$N(5; n) = T(5; n) = 0, \qquad n \geq 1 \tag{5}$

If P2 enneomino tiles $5 \times n$ rectangle then all rectangle cells should filled. Top left corner (marked with dot on image) may be covered in only six ways. First two are symmetric and merged in proof. All of them have untileable holes. Q.E.D.

$N(6; n) = 2 \times N(6; n - 6), \qquad n \geq 7 \tag{6}$

$N(15; n) = 16 \times N(15; n - 12), \qquad n \geq 13 \tag{7}$

See Also

P1 9-ominoP3 9-omino