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.


Y pentomino

Area: 5.

Perimeter: 12.

Size: 2x4.

Is rectangular: no.

Is convex: yes.

Holes: 0.

Order: 10.

Square order: 20.

Odd order: 45.

Prime rectangles: 40.

Smallest rectangle tilings

Smallest rectangle (5x10):

Smallest square (10x10):

Smallest odd rectangle (15x15):

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 \ h1-456-891011121314151617181920212223N>0
1-40
500
6-8000
90000
1001010P000350350C
110000000
1200000000
13000000000
140000296296P0000
150000089548954C00000093809380P16961696P
1600005050P00002003020030P0
17000000000011521152P00
18000000000000000
1900001351613516C0000133880133880P0000
200100100C0224224P242820242820C156156P003232P11070861107086C≥48554000≥48554000C≥1≥1C≥1≥1P≥1≥1C≥1≥1C≥1≥1C
21000027602760C0000≥1≥1P0000≥1≥1C0
220000000000≥1≥1P0000≥1≥1C00
23000058325832P0000≥1≥1P0000≥1≥1C000
240000602980602980C0000≥1≥1C0000≥1≥1C0005k
250000065911406591140C000000≥1≥1C≥1≥1C≥1≥1C≥1≥1P≥1≥1P≥1≥1C≥1≥1C≥1≥1C≥1≥1P≥1≥1Call
260000126088126088C0000≥1≥1C0000≥1≥1C0005k
270000496496P0000≥1≥1C0000≥1≥1C0005k
280000442978442978C0000≥1≥1C0000≥1≥1C0005k
2900002250699822506998C0000≥1≥1C0000≥1≥1C0005k
30010001000C01242012420P179340480179340480C2399223992P00≥68000≥68000P≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1Call
31000048437544843754C0000≥1≥1C0000≥1≥1C0005k
320000147924147924C0000≥1≥1C0000≥1≥1C0005k
3300002488544224885442C0000≥1≥1C0000≥1≥1C0005k
340000789989882789989882C0000≥1≥1C0000≥1≥1C0005k
35000004.89108913×10¹⁰4891089138C4850448504P00≥1≥1P≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1P≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1Call
360000172488058172488058C0000≥1≥1C0000≥1≥1C0005k
3700001391313813913138C0000≥1≥1C0000≥1≥1C0005k
3800001.18457646×10¹⁰1184576462C0000≥1≥1C0000≥1≥1C0005k
3900002.63870658×10¹¹26387065880C0000≥1≥1C0000≥1≥1C0005k
4001000010000C0579456579456C1.33548940×10¹²133548940522C1095153010951530C00≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1Call
4100005.82375181×10¹⁰5823751816C0000≥1≥1C0000≥1≥1C0005k
420000928325010928325010C0000≥1≥1C0000≥1≥1C0005k
4300005.05154848×10¹¹50515484824C0000≥1≥1C0000≥1≥1C0005k
4400008.53127100×10¹²853127100868C0000≥1≥1C0000≥1≥1C0005k
45000640640P3.65025521×10¹³3650255218632C7534153275341532P00≥1≥1P≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1Call
4600001.89958771×10¹²189958771098C0000≥1≥1C0000≥1≥1C0005k
4700005.17668000×10¹¹51766800024C0000≥1≥1C0000≥1≥1C0005k
4800002.00310134×10¹³2003101346584C0000≥1≥1C0000≥1≥1C0005k
4900002.69301190×10¹⁴26930119015566C0000≥1≥1C0000≥1≥1C0005k
500100000100000C02753229227532292C9.98522740×10¹⁴99852274075832C7.88568587×10¹⁰7885685870C6553665536P≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1Call
5100006.04698126×10¹³6046981261244C0000≥1≥1C0000≥1≥1C0005k
5200002.55608729×10¹³2556087298306C0000≥1≥1C0000≥1≥1C0005k
5300007.52541067×10¹⁴75254106745690C0000≥1≥1C0000≥1≥1C0005k
5400008.34898388×10¹⁵834898388157904C0000≥1≥1C0000≥1≥1C0005k
550007809678096P2.73343192×10¹⁶2733431927991712C6.27623714×10¹¹62762371436C20024002002400P≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1Call
5600001.89242891×10¹⁵189242891573924C0000≥1≥1C0000≥1≥1C0005k
5700001.15762741×10¹⁵115762741850480C0000≥1≥1C0000≥1≥1C0005k
5800002.71423891×10¹⁶2714238918610390C0000≥1≥1C0000≥1≥1C0005k
5900002.55239746×10¹⁷25523974639050546C0000≥1≥1C0000≥1≥1C0005k
60010000001000000C01.33167840×10¹⁰1331678408C7.48774482×10¹⁷74877448280303126C6.07504380×10¹³6075043806420C5692262456922624P≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1Call
6100005.85274226×10¹⁶5852742262997580C0000≥1≥1C0000≥1≥1C0005k
6200004.91212107×10¹⁶4912121079948382C0000≥1≥1C0000≥1≥1C0005k
6300009.48163614×10¹⁷94816361449973008C0000≥1≥1C0000≥1≥1C0005k
6400007.71652217×10¹⁸771652217309208422C0000≥1≥1C0000≥1≥1C0005k
6500071761287176128C2.05246057×10¹⁹2052460579702575446C4.68636448×10¹⁴46863644857792C1.64902008×10¹⁰1649020080P≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1Call
6600001.79593964×10¹⁸179593964264829212C0000≥1≥1C0000≥1≥1C0005k
6700001.98089015×10¹⁸198089015009165824C0000≥1≥1C0000≥1≥1C0005k
6800003.22847275×10¹⁹3228472753583779526C0000≥1≥1C0000≥1≥1C0005k
6900002.31186408×10²⁰23118640860020628390C0000≥1≥1C0000≥1≥1C0005k
7001000000010000000C06.49366215×10¹¹64936621544C5.62956329×10²⁰56295632921196943928C4.60042342×10¹⁶4600423424370562C5.16585390×10¹¹51658539068P≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1Call
7100005.48506546×10¹⁹5485065469124433186C0000≥1≥1C0000≥1≥1C0005k
7200007.66874511×10¹⁹7668745116916922912C0000≥1≥1C0000≥1≥1C0005k
7300001.07652404×10²¹107652404113825643450C0000≥1≥1C0000≥1≥1C0005k
7400006.87469458×10²¹687469458232923455740C0000≥1≥1C0000≥1≥1C0005k
75000551013048551013048C1.54507809×10²²1545078090243963120114C3.43980326×10¹⁷34398032698320668C1.72883751×10¹³1728837517320P≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1Call
7600001.67155350×10²¹167155350770225078780C0000≥1≥1C0000≥1≥1C0005k
7700002.87124858×10²¹287124858318866993920C0000≥1≥1C0000≥1≥1C0005k
7800003.52780553×10²²3527805537067101510262C0000≥1≥1C0000≥1≥1C0005k
7900002.03151826×10²³20315182626539081727170C0000≥1≥1C0000≥1≥1C0005k
800100000000100000000C03.18081201×10¹³3180812019640C4.24331934×10²³42433193492507540346022C3.41495393×10¹⁹3414953932729866368C6.07912516×10¹⁴60791251645044P≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1Call
8100005.09296950×10²²5092969508752021467788C0000≥1≥1C0000≥1≥1C0005k
8200001.04551400×10²³10455140061592409332088C0000≥1≥1C0000≥1≥1C0005k
8300001.13928605×10²⁴113928605166594411648768C0000≥1≥1C0000≥1≥1C0005k
8400005.97141363×10²⁴597141363575964204763794C0000≥1≥1C0000≥1≥1C0005k
850003.81474728×10¹¹38147472896C1.16613287×10²⁵1166132875327887767695324C2.52196392×10²⁰25219639252233427300C2.23890008×10¹⁶2238900081596314P≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1Call
8600001.55380398×10²⁴155380398368180595684220C0000≥1≥1C0000≥1≥1C0005k
8700003.71863273×10²⁴371863273015737470298292C0000≥1≥1C0000≥1≥1C0005k
8800003.63369444×10²⁵3633694447512431853114166C0000≥1≥1C0000≥1≥1C0005k
8900001.74724086×10²⁶17472408696306635652395242C0000≥1≥1C0000≥1≥1C0005k
9001.00000000×10¹⁰1000000000C01.56281224×10¹⁵156281224389490C3.20692506×10²⁶32069250620561200523983154C2.50728914×10²²2507289145823773296116C8.59316629×10¹⁷85931662961315302P≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1Call
9100004.75194067×10²⁵4751940675743734976656056C0000≥1≥1C0000≥1≥1C0005k
9200001.29633608×10²⁶12963360876918613954188508C0000≥1≥1C0000≥1≥1C0005k
9300001.14658024×10²⁷114658024444238225007664898C0000≥1≥1C0000≥1≥1C0005k
9400005.09232071×10²⁷509232071108809194370010770C0000≥1≥1C0000≥1≥1C0005k
950002.49089530×10¹³2490895301416C8.82558826×10²⁷882558826299585645352915388C1.85163999×10²³18516399941626698113328C3.42123297×10¹⁹3421232975806456186P≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1Call
9600001.45779854×10²⁷145779854593309837470100356C0000??0000??0005k
9700004.44146681×10²⁷444146681741890806437175084C0000??0000??0005k
9800003.58436604×10²⁸3584366047579138677396331926C0000??0000??0005k
9900001.47906427×10²⁹14790642747914977134929951990C0000??0000??0005k
10001.00000000×10¹¹10000000000C07.69538866×10¹⁶7695388668662110C2.43070780×10²⁹24307078072335322450857127430C1.83199211×10²⁵1831992116151385876258524C????????????????????????all
N>0x10kx5kall5k5k5k5kall5k5k5k5kall5k5k5k

Smallest prime reptiles

Smallest prime reptile (5Yx9):

Reptile tilings' solutions count (including symmetric)

polyomino \ n²10²11²12²13²
Y pentomino100000007629P≥46742129182700P≥20000P≥512P≥262144P

Smallest tori tilings

Smallest torus (2x5):

Smallest square torus (5x5):

Smallest odd torus (5x5):

Tori tilings' solutions count (including translations)

w \ h12345678
100
20000
3000000
400000000
5004040002602604040
6000000002320232000
7000000001401400000
800000000≥10000≥10000000000
900000000720720000000
100016016012012014201420≥10000≥10000≥10000≥10000≥10000≥10000≥10000≥10000
110000000033003300000000
1200000000≥10000≥10000000000
1300000000≥10000≥10000000000
1400000000≥10000≥10000000000
150064064000≥10000≥10000≥10000≥10000≥10000≥10000≥10000≥10000≥10000≥10000
1600000000≥10000≥10000000000
1700000000≥10000≥10000000000
1800000000≥10000≥10000000000
1900000000≥10000≥10000000000
200025602560480480≥10000≥10000≥10000≥10000≥10000≥10000≥10000≥10000≥10000≥10000
2100000000≥10000≥10000000000
2200000000≥10000≥10000000000
2300000000≥10000≥10000000000
2400000000≥10000≥10000000000
2500≥10000≥1000000≥10000≥10000≥10000≥10000≥10000≥10000≥10000≥10000≥10000≥10000
2600000000≥10000≥10000000000
2700000000≥10000≥10000000000
2800000000≥10000≥10000000000
2900000000≥10000≥10000000000
3000≥10000≥1000019201920≥10000≥10000≥10000≥10000≥10000≥10000≥10000≥10000??

Smallest Baiocchi figures

Smallest Baiocchi figure (area 20):

Smallest known Baiocchi figure without holes (area 40):

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 $A(w; h)$

$N(1; n) = T(1; n) = A(1; n) = 0 \tag{1}$

$N(2; n) = T(2; n) = A(2; n) = 0 \tag{2}$

$N(3; n) = T(3; n) = A(3; n) = 0 \tag{3}$

$N(4; n) = T(4; n) = A(4; n) = 0 \tag{4}$

$N(5; n) = 10 \times N(5; n - 10), \quad n \geq 11 \tag{5}$

$N(6; n) = T(6; n) = A(6; n) = 0 \tag{6}$

$N(7; n) = T(7; n) = A(7; n) = 0 \tag{7}$

$N(8; n) = T(8; n) = A(8; n) = 0 \tag{8}$

$G(N(5); x) = \frac{1}{1 - 10x^{10}} \tag{9}$

Attributions

  1. Smallest rectangle and smallest odd rectangle taken from http://www.cflmath.com/~reid/Polyomino/y5_rect.html
  2. Prime rectangles taken from http://www2.stetson.edu/~efriedma/order/index.html

See Also

X pentominoZ pentomino