POLYOMINO TILINGS

Polyomino Tilings

Select polyominoes for a set (currently 1 or 2), for which tilings should be shown.

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You may also see list of all polyomino sets for which data is available here.


Dominoes and L triomino

Prime rectangles: 7.

Smallest rectangle tilings

Smallest rectangle (2x4):

Smallest square (3x3):

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 \ h1234567N>0
10
2000
300088P
4066P4040P344344C
501616P140140P22242224C2127221272C
603636P584584C1487214872C260936260936C53439165343916C
709696P19841984C9535895358C≥1≥1C≥1≥1C≥1≥1C
80224224C76587658C617528617528C≥1≥1C≥1≥1C≥1≥1C?
90506506C2667626676C39563103956310C≥1≥1C≥1≥1C≥1≥1C?
10011661166C9901899018C≥1≥1C≥1≥1C≥1≥1C≥1≥1C?
11026242624C351080351080C≥1≥1C≥1≥1C≥1≥1C≥1≥1C?
12058565856C12810521281052C≥1≥1C≥1≥1C≥1≥1C≥1≥1C?
1301308813088C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C?
1402908829088C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C?
1506448264482C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C?
160142870142870C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C?
170316048316048C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C?
180698520698520C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C?
19015432321543232C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C?
20034076803407680C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C?
N>0xallallallallallall

Smallest prime reptiles

Smallest prime reptiles (2Ix2, 3Lx2):

Reptile tilings' solutions count (including symmetric)

polyomino \ n²
I domino06P584P617528C
L triomino034P28983P≥212429632C

Smallest common multiples

Smallest common multiple (area 6):

Common multiples' solutions count (excluding symmetric)

area6
solutions≥3P

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 $A(w; h)$

$N(1; h) = 0 \tag{1}$

See Also

Dominoes and I triominoDominoes and I tetromino