# 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.

## F hexomino¶

Area: 6.

Size: 2x4.

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 \ h1-2345678910111213N>0
1-20
300
4022P0
50000
60044C00
7000000
8044C001616C00
90088C0006464C0
10000000000
110000000000
12088C1616C06464C288288C256256C512512C35843584C72967296C81928192C
1300000000003891238912C00
140000000000125952125952C00?
15003232C00010241024C000193024193024C00?
1601616C00256256C0040964096C00458752458752C00?
17000000000017530881753088C00?
18006464C00040964096C00038338563833856C00?
19000000000070635527063552C00?
2003232C0010241024C003276832768C002267545622675456C00?
2100128128C0001638416384C0006324224063242240C00?
220000000000125124608125124608C00?
230000000000310321152310321152C00?
2406464C256256C040964096C8294482944C6553665536C262144262144C1525350415253504C5323161653231616C930086912930086912C2.05498777×10¹⁰2054987776C?
2500000000002.15364403×10¹⁰2153644032C00?
2600000000004.73969459×10¹⁰4739694592C00?
2700512512C000262144262144C0001.31950510×10¹¹13195051008C00?
280128128C001638416384C0020971522097152C003.43796613×10¹¹34379661312C00?
2900000000007.69429340×10¹¹76942934016C00?
300010241024C00010485761048576C0001.92346324×10¹²192346324992C00?
3100000000005.19077756×10¹²519077756928C00?
320256256C006553665536C001677721616777216C001.24361113×10¹³1243611136000C00?
330020482048C00041943044194304C0002.93979606×10¹³2939796062208C00?
3400000000007.69620810×10¹³7696208101376C00?
3500000000001.94859609×10¹⁴19485960962048C00?
360512512C40964096C0262144262144C2388787223887872C1677721616777216C134217728134217728C6.51416043×10¹¹65141604352C3.88377870×10¹²388377870336C4.61902928×10¹⁴46190292893696C1.07401547×10¹⁵107401547874304C?
3700000000001.15441238×10¹⁵115441238278144C00?
3800000000002.97501684×10¹⁵297501684924416C00?
390081928192C0006710886467108864C0007.26020585×10¹⁵726020585947136C00?
40010241024C0010485761048576C001.07374182×10¹⁰1073741824C001.76774465×10¹⁶1767744651395072C00?
4100000000004.50069129×10¹⁶4500691296976896C00?
42001638416384C000268435456268435456C0001.12656410×10¹⁷11265641067577344C00?
4300000000002.74152674×10¹⁷27415267428532224C00?
44020482048C0041943044194304C008.58993459×10¹⁰8589934592C006.84042446×10¹⁷68404244658520064C00?
45003276832768C0001.07374182×10¹⁰1073741824C0001.72750715×10¹⁸172750715230355456C00?
4600000000004.25594852×10¹⁸425594852944642048C00?
4700000000001.04907398×10¹⁹1049073988338188288C00?
48040964096C6553665536C01677721616777216C6.87970713×10¹⁰6879707136C4.29496729×10¹⁰4294967296C6.87194767×10¹¹68719476736C2.78217462×10¹⁵278217462120448C2.83360494×10¹⁶2833604941971456C2.63695495×10¹⁹2636954957323960320C5.61434559×10¹⁹5614345594878296064C?
4900000000006.57172888×10¹⁹6571728885157724160C00?
5000000000001.61834879×10²⁰16183487937273397248C00?
5100131072131072C0001.71798691×10¹¹17179869184C000≥1.84467440×10²⁰≥18446744073709551615C00?
52081928192C006710886467108864C005.49755813×10¹²549755813888C00≥1.84467440×10²⁰≥18446744073709551615C00?
530000000000≥1.84467440×10²⁰≥18446744073709551615C00?
5400262144262144C0006.87194767×10¹¹68719476736C000≥1.84467440×10²⁰≥18446744073709551615C00?
550000000000≥1.84467440×10²⁰≥18446744073709551615C00?
5601638416384C00268435456268435456C004.39804651×10¹³4398046511104C00≥1.84467440×10²⁰≥18446744073709551615C00?
5700524288524288C0002.74877906×10¹²274877906944C000≥1.84467440×10²⁰≥18446744073709551615C00?
580000000000≥1.84467440×10²⁰≥18446744073709551615C00?
590000000000≥1.84467440×10²⁰≥18446744073709551615C00?
6003276832768C10485761048576C01.07374182×10¹⁰1073741824C1.98135565×10¹³1981355655168C1.09951162×10¹³1099511627776C3.51843720×10¹⁴35184372088832C1.18825925×10¹⁹1188259258064437248C≥1.84467440×10²⁰≥18446744073709551615C≥1.84467440×10²⁰≥18446744073709551615C≥1.84467440×10²⁰≥18446744073709551615C?
N>0x4k3kx4k12k3k4k12k12kall?

## Smallest prime reptiles¶

Smallest prime reptile (6Fx8):

## Reptile tilings' solutions count (including symmetric)¶

polyomino \ n²
F hexomino1000000≥3200000P≥1P

## 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 $A(w; h)$

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

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

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

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

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

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

$N(7; n) = 288 \times N(7; n - 12), \qquad n \geq 13 \tag{7}$

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

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

$N(10; n) = 4384 \times N(10; n - 12) - 483584 \times N(10; n - 24) + 3604480 \times N(10; n - 36) + 100663296 \times N(10; n - 48), \qquad n \geq 49 \tag{10}$

$N(11; n) = 7296 \times N(11; n - 12), \qquad n \geq 13 \tag{11}$

$N(12; n) = 8 \times N(12; n - 3) + 20 \times N(12; n - 4) - 64 \times N(12; n - 8), \qquad n \geq 9 \tag{12}$