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.


J1 octomino

Area: 8.

Size: 3x4.

Holes: 0.

Order: 2.

Square order: 2.

Prime rectangles: ≥ 1.

Smallest rectangle tilings

Smallest rectangle and smallest square (4x4):

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 \ h1-345-789-1112N>0
1-30
4044P
5-7000
801616C0256256C
9-1100000
1206464C040964096C0262144262144C
13000000?
14000000?
15000000?
160256256C06553665536C01677721616777216C?
17000000?
18000000?
19000000?
20010241024C010485761048576C01.07374182×10¹⁰1073741824C?
21000000?
22000000?
23000000?
24040964096C01677721616777216C06.87194767×10¹¹68719476736C?
25000000?
26000000?
27000000?
2801638416384C0268435456268435456C04.39804651×10¹³4398046511104C?
29000000?
30000000?
31000000?
3206553665536C04.29496729×10¹⁰4294967296C02.81474976×10¹⁵281474976710656C?
N>0x4kx4kx4k

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 $A(w; h)$

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

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

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

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

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

$N(12; n) = 64 \times N(12; n - 4), \qquad n \geq 5 \tag{6}$

See Also

I2 octominoL1 octomino