# 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.

## G 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 - strongly prime rectangle (which cannot be divided into two or more number of rectangles tileable by this set).

Green number - weakly prime rectangle (which cannot be divided into two rectangles tileable by this set, but which can be divided into three or more rectangles).

Purple number - prime rectangle (unknown if weakly or strongly prime).

Red number - 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 \ h
1-3
4
5-7
8
9-11
12
13-15
16
17
18
19
20
1-3
0
4
0
4
5-7
0
0
0
8
0
16
0
256
9-11
0
0
0
0
0
12
0
64
0
4096
0
262144
13-15
0
0
0
0
0
0
0
16
0
256
0
65536
0
16777216
0
≥1
17
0
0
0
0
0
0
0
0
?
18
0
0
0
0
0
0
0
0
?
?
19
0
0
0
0
0
0
0
0
?
?
?
20
0
≥1
0
≥1
0
≥1
0
≥1
?
?
?
≥1
N>0
x
4k
x
4k
x
4k
x
4k
?
?
?
?

## Smallest prime reptiles¶

Smallest prime reptile (8Gx4):

polyomino \ n²
G octomino
1
0
0
65536
0
0
0
≥4294967296

## 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 $A(w; h)$

$N(w; h) = T(w; h) = A(w; h) = 0, \qquad w \not\equiv 0\pmod{4} \tag{1}$

$T(4n; 4m) = 1, \qquad n,m \geq 1 \tag{2}$

$N(4n; 4m) = 4^{nm}, \qquad n,m \geq 1 \tag{3}$

$G(T; x; y) = \frac{x^4y^4}{(1-x^4)(1-y^4)} \tag{4}$

$A(4n; 4m) = \sqrt[8]{2}, \qquad n,m \geq 1 \tag{5}$

$G(A; x; y) = \sqrt[8]{2}G(T; x; y) = \frac{\sqrt[8]{2}\,x^4y^4}{(1-x^4)(1-y^4)} \tag{6}$