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.


A hexomino

Area: 6.

Size: 3x3.

Holes: 0.

Order: 2.

Square order: 24.

Prime rectangles: ≥ 1.

Smallest rectangle tilings

Smallest rectangle (3x4):

Smallest square (12x12):

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 \ h1-2345678910111213141516N>0
1-20
300
4022P0
50000
60044C00
7000000
8044C001616C00
90088C0006464C0
10000000000
110000000000
12088C1616C06464C256256C256256C512512C30723072C61446144C81928192C
1300000000003276832768C0
1400000000009830498304C00
15003232C00010241024C000163840163840C000
1601616C00256256C0040964096C00393216393216C001782579217825792C0
17000000000013107201310720C0000?
18006464C00040964096C00028835842883584C000612368384612368384C?
19000000000057671685767168C0000?
2003232C0010241024C003276832768C001677721616777216C001.45961779×10¹⁰1459617792C0?
2100128128C0001638416384C0004404019244040192C0001.79851755×10¹¹17985175552C?
2200000000009227468892274688C0000?
230000000000226492416226492416C0000?
2406464C256256C040964096C6553665536C6553665536C262144262144C94371849437184C3774873637748736C620756992620756992C1.07374182×10¹⁰1073741824C9.66367641×10¹⁰9663676416C1.17037858×10¹²117037858816C5.84115552×10¹²584115552256C?
2500000000001.44284057×10¹⁰1442840576C0000?
2600000000003.28833433×10¹⁰3288334336C0000?
2700512512C000262144262144C0008.58993459×10¹⁰8589934592C0002.10281598×10¹⁴21028159881216C?
280128128C001638416384C0020971522097152C002.14748364×10¹¹21474836480C009.89560464×10¹³9895604649984C0?
2900000000004.93921239×10¹¹49392123904C0000?
300010241024C00010485761048576C0001.21332826×10¹²121332826112C0007.70207895×10¹⁵770207895257088C?
3100000000003.09237645×10¹²309237645312C0000?
320256256C006553665536C001677721616777216C007.38734374×10¹²738734374912C008.49922488×10¹⁵849922488270848C0?
330020482048C00041943044194304C0001.76093659×10¹³1760936591360C0002.69864133×10¹⁷26986413392134144C?
3400000000004.41522638×10¹³4415226380288C0000?
3500000000001.08576773×10¹⁴10857677324288C0000?
360512512C40964096C0262144262144C1677721616777216C1677721616777216C134217728134217728C2.89910292×10¹¹28991029248C2.31928233×10¹²231928233984C2.59072427×10¹⁴25907242729472C3.51843720×10¹⁴35184372088832C9.49978046×10¹⁵949978046398464C7.24270299×10¹⁷72427029944860672C9.21337967×10¹⁸921337967518154752C?
3700000000006.34967965×10¹⁴63496796504064C0000?
3800000000001.57505040×10¹⁵157505040678912C0000?
390081928192C0006710886467108864C0003.80980779×10¹⁵380980779024384C000≥1.84467440×10²⁰≥18446744073709551615C?
40010241024C0010485761048576C001.07374182×10¹⁰1073741824C009.22490255×10¹⁵922490255704064C006.14206546×10¹⁹6142065466803224576C0?
4100000000002.27598906×10¹⁶2275989069496320C0000?
42001638416384C000268435456268435456C0005.56792688×10¹⁶5567926883057664C000≥1.84467440×10²⁰≥18446744073709551615C?
4300000000001.34756145×10¹⁷13475614510022656C0000?
44020482048C0041943044194304C008.58993459×10¹⁰8589934592C003.29677566×10¹⁷32967756647235584C00≥1.84467440×10²⁰≥18446744073709551615C0?
45003276832768C0001.07374182×10¹⁰1073741824C0008.09592401×10¹⁷80959240176402432C000≥1.84467440×10²⁰≥18446744073709551615C?
4600000000001.96891746×10¹⁸196891746209103872C0000?
4700000000004.79351885×10¹⁸479351885338247168C0000?
48040964096C6553665536C01677721616777216C4.29496729×10¹⁰4294967296C4.29496729×10¹⁰4294967296C6.87194767×10¹¹68719476736C8.90604418×10¹⁴89060441849856C1.42496706×10¹⁶1424967069597696C1.17515802×10¹⁹1175158027766988800C1.15292150×10¹⁹1152921504606846976C9.33866418×10²⁰93386641873154605056C≥1.84467440×10²⁰≥18446744073709551615C≥1.84467440×10²⁰≥18446744073709551615C?
4900000000002.87048181×10¹⁹2870481812495269888C0000?
5000000000006.98508302×10¹⁹6985083022051639296C0000?
5100131072131072C0001.71798691×10¹¹17179869184C0001.70708943×10²⁰17070894387547865088C000≥1.84467440×10²⁰≥18446744073709551615C?
52081928192C006710886467108864C005.49755813×10¹²549755813888C004.17663829×10²⁰41766382944233979904C00≥1.84467440×10²⁰≥18446744073709551615C0?
5300000000001.01808373×10²¹101808373176337432576C0000?
5400262144262144C0006.87194767×10¹¹68719476736C0002.48328483×10²¹248328483453209149440C000≥1.84467440×10²⁰≥18446744073709551615C?
5500000000006.07265373×10²¹607265373754637680640C0000?
5601638416384C00268435456268435456C004.39804651×10¹³4398046511104C001.48272911×10²²1482729112518443139072C00≥1.84467440×10²⁰≥18446744073709551615C0?
5700524288524288C0002.74877906×10¹²274877906944C0003.61556183×10²²3615561838447072116736C000≥1.84467440×10²⁰≥18446744073709551615C?
5800000000008.83137872×10²²8831378725288447836160C0000?
5900000000002.15780788×10²³21578078880221748002816C0000?
6003276832768C10485761048576C01.07374182×10¹⁰1073741824C1.09951162×10¹³1099511627776C1.09951162×10¹³1099511627776C3.51843720×10¹⁴35184372088832C2.73593677×10¹⁸273593677362757632C8.75499767×10¹⁹8754997675608244224C5.26481605×10²³52648160507871667159040C3.77789318×10²³37778931862957161709568C9.18028044×10²⁵9180280442698590295425024C≥1.84467440×10²⁰≥18446744073709551615C≥1.84467440×10²⁰≥18446744073709551615C?
N>0x4k3kx4k12k3k4k12k12kall12k12k4k3k

Reptile tilings' solutions count (including symmetric)

polyomino \ n²10²11²12²13²
A hexomino1000000???≥576460752303423488P≥131744580139097362171691008P≥9671406556917033397649408P

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 $A(w; h)$

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

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

$N(3; n) = 2 \times N(3; n - 4), \qquad n \geq 5 \tag{3}$

$N(4; n) = 2 \times N(4; n - 3), \qquad n \geq 4 \tag{4}$

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

$N(6; n) = 4 \times N(6; n - 4), \qquad n \geq 5 \tag{6}$

$N(7; n) = 256 \times N(7; n - 12), \qquad n \geq 13 \tag{7}$

$N(8; n) = 4 \times N(8; n - 3), \qquad n \geq 4 \tag{8}$

$N(9; n) = 8 \times N(9; n - 4), \qquad n \geq 5 \tag{9}$

$N(10; n) = 3072 \times N(10; n - 12), \qquad n \geq 13 \tag{10}$

$N(11; n) = 6144 \times N(11; n - 12), \qquad n \geq 13 \tag{11}$

$N(12; n) = 8 \times N(12; n - 3) + 16 \times N(12; n - 4), \qquad n \geq 5 \tag{12}$

$N(13; n) = 32768 \times N(13; n - 12), \qquad n \geq 13 \tag{13}$

$N(14; n) = 98304 \times N(14; n - 12), \qquad n \geq 13 \tag{14}$

$N(15; n) = 128 \times N(15; n - 4) - 6144 \times N(15; n - 8) + 262144 \times N(15; n - 12) - 5242880 \times N(15; n - 16) + 83886080 \times N(15; n - 20) - 536870912 \times N(15; n - 24), \qquad n \geq 25 \tag{15}$

$N(16; n) = 48 \times N(16; n - 3) - 768 \times N(16; n - 6) + 4096 \times N(16; n - 9) + 327680 \times N(16; n - 12) - 4194304 \times N(16; n - 15) + 25165824 \times N(16; n - 18), \qquad n \geq 19 \tag{16}$

$G(N(3); x) = \frac{1}{1 - 2x^4} \tag{17}$

$G(N(4); x) = \frac{1}{1 - 2x^3} \tag{18}$

$G(N(6); x) = \frac{1}{1 - 4x^4} \tag{19}$

$G(N(7); x) = \frac{1}{1 - 256x^{12}} \tag{20}$

$G(N(8); x) = \frac{1}{1 - 4x^3} \tag{21}$

$G(N(9); x) = \frac{1}{1 - 8x^4} \tag{22}$

$G(N(10); x) = \frac{1}{1 - 3072x^{12}} \tag{23}$

$G(N(11); x) = \frac{1}{1 - 6144x^{12}} \tag{24}$

$G(N(12); x) = \frac{1}{1 - 8x^3 - 16x^4} \tag{25}$

See Also

Z pentominoB hexomino