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.


B1 9-omino

Area: 9.

Size: 3x5.

Is rectangular: no.

Is convex: yes.

Holes: 0.

Order: 4.

Square order: 4.

Prime rectangles: ≥ 21.

Smallest rectangle tilings

Smallest rectangle and smallest square (6x6):

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 \ h1-567-111213-17181920-21222324N>0
1-50
6022P
7-11000
12066C08686C
13-1700000
1801818C012381238C08617286172C
19000000000
20-210000000000
22000000000000
2300000000000000
2405454C01777817778C059645905964590C00000002.00789374×10¹⁰2007893740C
2500000816816P00000000?
26000000000000000?
270000000000000000?
28000000000000000?
29000000000000000?
300162162C0255118255118C0411969510411969510C00000006.74277710×10¹²674277710782C?
3100000136272136272C00000000?
32000007272P00000000?
330000000000000065377286537728?
340000084488448P00000000?
35000000000000000?
360486486C036605743660574C02.84463831×10¹¹28446383104C00000640224640224P2.26288166×10¹⁵226288166103910C?
37000001552502415525024C00000000?
38000001853618536C00000000?
39000000000000007.12824877×10¹⁰7128248776C?
400000018001921800192C00000000?
410000030723072P00000000?
42014581458C05252407852524078C01.96423403×10¹³1964234037658C00000007.59476876×10¹⁷75947687668024428C?
43000001.51266577×10¹⁰1512665776C00000000?
440000030631123063112C00000000?
450000012801280P00000004.93099799×10¹³4930997994976C?
4600000245804544245804544C00000000?
470000011320321132032C00000000?
48043744374C0753658970753658970C01.35634653×10¹⁵135634653256334C0000000≥1.84467440×10²⁰≥18446744073709551615C?
49000001.35584043×10¹²135584043680C00000000?
5000000397533112397533112C00000000?
5100000543808543808C00000002.76933023×10¹⁶2769330232932192C?
52000002.73535599×10¹¹27353559936C00000000?
5300000209131008209131008C00000000?
5401312213122C01.08142644×10¹¹10814264438C09.36602473×10¹⁶9366024733787484C000005.46481101×10¹²546481101568P≥1.84467440×10²⁰≥18446744073709551615C?
55000001.15426735×10¹⁴11542673502512C00000000?
56000004.43257137×10¹¹44325713784C00000000?
5700000114617152114617152C00000001.37743659×10¹⁹1377436599543259688C?
58000002.70544613×10¹³2705446132992C00000000?
59000002.82401551×10¹¹28240155136C00000000?
6003936639366C01.55175389×10¹²155175389958C06.46761789×10¹⁸646761789846709110C0000000≥1.84467440×10²⁰≥18446744073709551615C?
61000009.48994231×10¹⁵948994231590976C00000000?
62000004.46607198×10¹³4466071989104C00000000?
63000001.73490235×10¹¹17349023552C0000000≥1.84467440×10²⁰≥18446744073709551615C?
64000002.48064579×10¹⁵248064579101888C00000000?
65000003.19621357×10¹³3196213572992C00000000?
660118098118098C02.22664547×10¹³2226645475942C04.46617978×10²⁰44661797843698814078C0000000≥1.84467440×10²⁰≥18446744073709551615C?
67000007.60866669×10¹⁷76086666908941936C00000000?
68000004.19209730×10¹⁵419209730555368C00000000?
69000002.16279723×10¹³2162797237312C0000000≥1.84467440×10²⁰≥18446744073709551615C?
70000002.15913351×10¹⁷21591335117897184C00000000?
71000003.23703866×10¹⁵323703866937856C00000000?
720354294354294C03.19507298×10¹⁴31950729878818C03.08410992×10²²3084109929905246681232C9.91368252×10¹¹99136825216P0007.42711410×10¹⁷74271141091376432C≥1.84467440×10²⁰≥18446744073709551615C?
73000005.98587759×10¹⁹5985877590700977168C00000000?
74000003.73776054×10¹⁷37377605451343656C00000000?
75000002.37566700×10¹⁵237566700626560C0000000≥1.84467440×10²⁰≥18446744073709551615C?
76000001.81008680×10¹⁹1810086809902998560C00000000?
77000003.04138426×10¹⁷30413842656094208C00000000?
78010628821062882C04.58470518×10¹⁵458470518380862C02.12973057×10²⁴212973057958974852008562C0000000≥1.84467440×10²⁰≥18446744073709551615C?
79000004.64017230×10²¹464017230695874740128C00000000?
80000003.20687728×10¹⁹3206877288018903968C00000000?
81000002.38949670×10¹⁷23894967059608896C0000000≥1.84467440×10²⁰≥18446744073709551615C?
82000001.47551818×10²¹147551818872624492128C00000000?
83000002.70930382×10¹⁹2709303824389346176C00000000?
84031886463188646C06.57873840×10¹⁶6578738403025550C01.47068696×10²⁶14706869677023061257804870C0000000≥1.84467440×10²⁰≥18446744073709551615C?
85000003.55473649×10²³35547364934283738923104C00000000?
86000002.67130541×10²¹267130541039568739544C00000000?
87000002.25431502×10¹⁹2254315020972681152C0000000≥1.84467440×10²⁰≥18446744073709551615C?
88000001.17713692×10²³11771369285161773793632C00000000?
89000002.32051298×10²¹232051298852115635072C00000000?
90095659389565938C09.44004745×10¹⁷94400474556742142C01.01558508×10²⁸1015585089137135091110970492C1.74762389×10¹⁵174762389421312P000≥1.84467440×10²⁰≥18446744073709551615C≥1.84467440×10²⁰≥18446744073709551615C?
91000002.69702034×10²⁵2697020344006171143859360C00000000?
92000002.17420472×10²³21742047237390215025952C00000000?
93000002.02690408×10²¹202690408598915282432C0000000≥1.84467440×10²⁰≥18446744073709551615C?
94000009.23284549×10²⁴923284549704085963922112C00000000?
95000001.92907613×10²³19290761343557036126208C00000000?
9602869781428697814C01.35458405×10¹⁹1354584055561251402C07.01314324×10²⁹70131432489564572811636989214C0000000≥1.84467440×10²⁰≥18446744073709551615C?
97000002.02987585×10²⁷202987585283666832671640400C00000000?
98000001.73714746×10²⁵1737147465760954456468832C00000000?
99000001.75615642×10²³17561564257736121198336C0000000≥1.84467440×10²⁰≥18446744073709551615C?
100000007.14362526×10²⁶71436252691425881997719360C00000000?
101000001.56684183×10²⁵1566841839866303437884032C00000000?
10208609344286093442C01.94373868×10²⁰19437386886455417446C04.84294250×10³¹4842942505826657256366873062838C0000000≥1.84467440×10²⁰≥18446744073709551615C?
103000001.51742228×10²⁹15174222865180242457896948688C00000000?
104000001.36724499×10²⁷136724499919622530846204056C00000000?
105000001.47791181×10²⁵1477911814252760197382336C0000000≥1.84467440×10²⁰≥18446744073709551615C?
106000005.46584173×10²⁸5465841732630911234776534336C00000000?
107000001.24935208×10²⁷124935208457153136317851264C00000000?
1080258280326258280326C02.78913724×10²¹278913724130578178614C03.34430634×10³³334430634742756244192391914964512C3.84296728×10¹⁸384296728343941504P000≥1.84467440×10²⁰≥18446744073709551615C≥1.84467440×10²⁰≥18446744073709551615C?
109000001.12777443×10³¹1127774436645357328002790506880C00000000?
110000001.06287860×10²⁹10628786035193980624673121240C00000000?
111000001.21514058×10²⁷121514058403582613621053440C0000000≥1.84467440×10²⁰≥18446744073709551615C?
112000004.14356587×10³⁰414356587546908372522449894464C00000000?
113000009.81440105×10²⁸9814401055248808982001370368C00000000?
1140774840978774840978C04.00222908×10²²4002229081364023599990C02.30941978×10³⁵23094197834545322626328119890709866C0000000≥1.84467440×10²⁰≥18446744073709551615C?
115000008.33985181×10³²83398518134791988802019744795664C00000000?
116000008.17792299×10³⁰817792299821499598424463229240C00000000?
117000009.80409679×10²⁸9804096790845172919936104832C0000000≥1.84467440×10²⁰≥18446744073709551615C?
118000003.11680223×10³²31168022390380170987496667273120C00000000?
119000007.61593463×10³⁰761593463794839249113828111744C00000000?
12002.32452293×10¹⁰2324522934C05.74293677×10²³57429367784344048193618C01.59477628×10³⁷1594776283176596155907881858249353310C0000000≥1.84467440×10²⁰≥18446744073709551615C?
121000006.14033189×10³⁴6140331891165022322057722465521504C00000000?
122000006.23774089×10³²62377408977733536845463669770216C00000000?
123000007.78854055×10³⁰778854055473085255160267406656C0000000≥1.84467440×10²⁰≥18446744073709551615C?
124000002.32898061×10³⁴2328980610724125060476123857857504C00000000?
125000005.85001869×10³²58500186972968151805717170736512C00000000?
12606.97356880×10¹⁰6973568802C08.24073878×10²⁴824073878264498926355822C01.10127731×10³⁹110127731311724911620082617914996921052C≥1.84467440×10²⁰≥18446744073709551615P000≥1.84467440×10²⁰≥18446744073709551615C≥1.84467440×10²⁰≥18446744073709551615C?
127000004.50350460×10³⁶450350460663543029197060070490368656C00000000?
128000004.72276287×10³⁴4722762874494000010210590534229368C00000000?
129000006.10817756×10³²61081775640375751821918227700480C0000000≥1.84467440×10²⁰≥18446744073709551615C?
130000001.73040228×10³⁶173040228885210226772518725336650016C00000000?
131000004.45518173×10³⁴4455181735003743060456850959959552C00000000?
13202.09207064×10¹¹20920706406C01.18249216×10²⁶11824921692518724118732926C07.60490234×10⁴⁰7604902340848872217266306358013907480294C0000000≥1.84467440×10²⁰≥18446744073709551615C?
133000003.29172923×10³⁸32917292318589141819513453986693688160C00000000?
134000003.55303912×10³⁶355303912570235383687971789563503888C00000000?
135000004.73884898×10³⁴4738848984272194785029234535335744C0000000≥1.84467440×10²⁰≥18446744073709551615C?
136000001.27932038×10³⁸12793203866382792748128165597014506176C00000000?
137000003.36823627×10³⁶336823627086897302486426144378022656C00000000?
13806.27621192×10¹¹62762119218C01.69679902×10²⁷169679902847827658864065870C05.25158750×10⁴²525158750240746623247277752986947425667662C0000000≥1.84467440×10²⁰≥18446744073709551615C?
139000002.39867566×10⁴⁰2398675669105273480404940992145558146800C00000000?
140000002.65830930×10³⁸26583093090837691322021193485332901848C00000000?
141000003.64297673×10³⁶364297673018576292112138267719168384C0000000≥1.84467440×10²⁰≥18446744073709551615C?
142000009.41738137×10³⁹941738137126204990050641822437522849664C00000000?
143000002.53054594×10³⁸25305459458839865892466541009142337664C00000000?
14401.88286357×10¹²188286357654C02.43479580×10²⁸2434795807371589988317167034C03.62649916×10⁴⁴36264991607829481431100875597002734481988464C≥1.84467440×10²⁰≥18446744073709551615C000≥1.84467440×10²⁰≥18446744073709551615C≥1.84467440×10²⁰≥18446744073709551615C?
145000001.74312248×10⁴²174312248073595764449184017109089114143344C00000000?
146000001.97931075×10⁴⁰1979310759954108366406555919071009313272C00000000?
147000002.77868746×10³⁸27786874689728266880027433841964402304C0000000≥1.84467440×10²⁰≥18446744073709551615C?
148000006.90591511×10⁴¹69059151142165178812860455590316924400640C00000000?
149000001.89087318×10⁴⁰1890873185484830220923210865720231934976C00000000?
15005.64859072×10¹²564859072962C03.49377301×10²⁹34937730198020147781521561046C02.50428967×10⁴⁶2504289676742771801304766052570378112140229954C0000000≥1.84467440×10²⁰≥18446744073709551615C?
151000001.26359729×10⁴⁴12635972970300603253189582719704830129759104C00000000?
152000001.46749033×10⁴²146749033289406673174988160755947602811536C00000000?
153000002.10519553×10⁴⁰2105195535666413364336842918102293909568C0000000≥1.84467440×10²⁰≥18446744073709551615C?
154000005.04705392×10⁴³5047053928313325929167502174474230338493568C00000000?
155000001.40619039×10⁴²140619039186269911423151010972732380429568C00000000?
15601.69457721×10¹³1694577218886C05.01333620×10³⁰501333620293298293380357951974C01.72934462×10⁴⁸172934462988897599721920859661810829631388039062C0000000≥1.84467440×10²⁰≥18446744073709551615C?
157000009.13930928×10⁴⁵913930928560522587907114138961630350160738336C00000000?
158000001.08392256×10⁴⁴10839225694754149845017525814499990713282216C00000000?
159000001.58562280×10⁴²158562280168146617711935381962186949092288C0000000≥1.84467440×10²⁰≥18446744073709551615C?
160000003.67736505×10⁴⁵367736505463410477999427780624922042915692512C00000000?
161000001.04137111×10⁴⁴10413711185642645857465256764547320601270784C00000000?
16205.08373165×10¹³5083731656658C07.19381019×10³¹7193810190379015151788059505670C01.19420404×10⁵⁰11942040478011855361879288240100581717972673909948C≥1.84467440×10²⁰≥18446744073709551615C0≥1.84467440×10²⁰≥18446744073709551615P≥1.84467440×10²⁰≥18446744073709551615C≥1.84467440×10²⁰≥18446744073709551615C?
163000006.59672361×10⁴⁷65967236108382801716021654235835290781395590784C00000000?
164000007.97919699×10⁴⁵797919699389097231455861787464821354749201856C00000000?
165000001.18817559×10⁴⁴11881755994338169154294288660050533162021184C0000000≥1.84467440×10²⁰≥18446744073709551615C?
166000002.67208803×10⁴⁷26720880312854264479193697653510881855999271296C00000000?
167000007.68338105×10⁴⁵768338105216244558874536545881492312983089408C00000000?
16801.52511949×10¹⁴15251194969974C01.03226480×10³³103226480429122621630848826931970C08.24661140×10⁵¹824661140029384615052280314644518994236586089322510C0000000≥1.84467440×10²⁰≥18446744073709551615C?
169000004.75257309×10⁴⁹4752573099669812984944692759853105266716342536176C00000000?
170000005.85608923×10⁴⁷58560892371553956865202228828438198521804270016C00000000?
171000008.86339603×10⁴⁵886339603643109827481652599843729640203130240C0000000≥1.84467440×10²⁰≥18446744073709551615C?
172000001.93684172×10⁴⁹1936841721883229852889655558712554528441478070464C00000000?
173000005.65012553×10⁴⁷56501255345216576942216318135901387749793673984C00000000?
17404.57535849×10¹⁴45753584909922C01.48123261×10³⁴1481232614027479428179484046851358C05.69472192×10⁵³56947219284921456338206060582621221172210066745579014C0000000≥1.84467440×10²⁰≥18446744073709551615C?
175000003.41807135×10⁵¹341807135013777761024358046133186800145315789045328C00000000?
176000004.28619364×10⁴⁹4286193640747182654336604710448558862155999794344C00000000?
177000006.58538646×10⁴⁷65853864639348806768195911870078236590981488256C0000000≥1.84467440×10²⁰≥18446744073709551615C?
178000001.40076535×10⁵¹140076535852797814608491743694374701340878218141952C00000000?
179000004.14257169×10⁴⁹4142571694427134011156589169446961113700299028736C00000000?
18001.37260754×10¹⁵137260754729766C02.12547211×10³⁵21254721164457848890142583086051566C03.93250710×10⁵⁵3932507102526821501740806210352432609511543260070670784C≥1.84467440×10²⁰≥18446744073709551615C0≥1.84467440×10²⁰≥18446744073709551615P≥1.84467440×10²⁰≥18446744073709551615C≥1.84467440×10²⁰≥18446744073709551615C?
181000002.45439068×10⁵³24543906808199234055763869627524061694531098127991392C00000000?
182000003.12940491×10⁵¹312940491465343304168067543732165212362597967380872C00000000?
183000004.87542247×10⁴⁹4875422473461200982069969319526581951256449414144C0000000≥1.84467440×10²⁰≥18446744073709551615C?
184000001.01099770×10⁵³10109977025736224168770160568739018625529539849469952C00000000?
185000003.02910438×10⁵¹302910438376152632762544661991971343452855988954112C00000000?
18604.11782264×10¹⁵411782264189298C03.04991375×10³⁶304991375214372366867913492772289054C02.71560443×10⁵⁷271560443593909534850899549056277153117040412543732108986C0000000≥1.84467440×10²⁰≥18446744073709551615C?
187000001.75982050×10⁵⁵1759820509775373272876172426277147895181011456225651056C00000000?
188000002.27967623×10⁵³22796762317740042444114729784666999323096917016119464C00000000?
189000003.59793483×10⁵¹359793483123856817643379855968449268874682236990464C0000000≥1.84467440×10²⁰≥18446744073709551615C?
190000007.28321794×10⁵⁴728321794609108437496108698316891638963792811899261888C00000000?
191000002.20952696×10⁵³22095269611793272763718424518259459566829629457668096C00000000?
19201.23534679×10¹⁶1235346792567894C04.37642715×10³⁷4376427159839213057730617377212707498C01.87526869×10⁵⁹18752686940512373779692820365409834915558106132121510490414C0000000≥1.84467440×10²⁰≥18446744073709551615C?
193000001.26008950×10⁵⁷126008950996941358607920013366548969607611903807478324512C00000000?
194000001.65725499×10⁵⁵1657254994817974600759374885851512976364084464348016856C00000000?
195000002.64753370×10⁵³26475337029568246355868139100216265419144328465449280C0000000≥1.84467440×10²⁰≥18446744073709551615C?
196000005.23782031×10⁵⁶52378203181895162968697986862801062838525166396098437056C00000000?
197000001.60812338×10⁵⁵1608123384187181436943364479018774247421902926610703360C00000000?
19803.70604037×10¹⁶3706040377703682C06.27988731×10³⁸62798873164732069683000052852402318022C01.29497235×10⁶¹1294972356895211904346781363110267120261880555738103743724380C≥1.84467440×10²⁰≥18446744073709551615C0≥1.84467440×10²⁰≥18446744073709551615P≥1.84467440×10²⁰≥18446744073709551615C≥1.84467440×10²⁰≥18446744073709551615C?
199000009.01123128×10⁵⁸9011231287842762041867178552316490079719481520054483559920C00000000?
200000001.20249504×10⁵⁷120249504473483848676125983403336881725222275334904186296C00000000?
201000001.94309914×10⁵⁵1943099140648549136104801130059719527753191497204220352C0000000≥1.84467440×10²⁰≥18446744073709551615C?
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319000009.24489929×10⁹⁵92448992992214696387174649128908516843559549485669205068888982153152290226496380482601554267728C??0????0?
320000001.53228156×10⁹⁴1532281569855491924070871007724730344731280422441973197393512383154677325694495365155181026824C??0????0?
321000002.88596755×10⁹²28859675507501682033064840358788640300130852755040381446034897234251176260977329522910327552C??0???????
322000004.03673913×10⁹⁵40367391328022016117223256697261028416365811523221166264278756245170559060407771419973619669376C??0????0?
323000001.49792830×10⁹⁴1497928304678525114687643875768374369182778665712976385811333400651202093093925080620364641536C??0????0?
32403.87664913×10²⁶38766491335360039793593446C01.23431761×10⁶³123431761973598199187664552149329168384313854962353580256381246C05.43747647×10⁹⁹543747647148367150500912106387497508249501074047289330021325569432362459350931316879950473673371008C??0???????
N>0x6kx6kxall?x??3k

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 $A(w; h)$

$N(6; n) = 3 \times N(6; n - 6), \qquad n \geq 7 \tag{1}$

$N(12; n) = 29 \times N(12; n - 6) - 272 \times N(12; n - 12) + 988 \times N(12; n - 18) - 1515 \times N(12; n - 24) + 819 \times N(12; n - 30), \qquad n \geq 31 \tag{2}$

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

See Also

A 9-ominoI 9-omino