# 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 (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
13-15000000x
160256256C06553665536C01677721616777216C4k
N>0x4kx4kx4k

## Smallest prime reptiles¶

Smallest prime reptile (8Gx4):

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

polyomino \ n²
G octomino10065536P000≥4294967296P

## 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 $A(w; h)$

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

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

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

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

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

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