# 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.

## L hexomino¶

Area: 6.

Perimeter: 14.

Size: 2x5.

Is rectangular: no.

Is convex: yes.

Holes: 0.

Order: 2.

Square order: 6.

Odd order: 21.

Prime rectangles: 8.

## Smallest rectangle tilings¶

Smallest rectangle (2x6):

Smallest square (6x6):

Smallest odd rectangle (9x14):

## 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
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
N>0
1
0
2
0
0
3
0
0
0
4
0
0
0
0
5
0
0
0
0
0
6
0
2
0
4
0
16
7
0
0
0
0
0
0
0
8
0
0
0
0
0
48
0
0
9
0
0
0
0
0
0
0
0
0
10
0
0
0
0
0
128
0
0
0
0
11
0
0
0
0
0
0
0
0
0
0
0
12
0
4
0
16
0
384
300
3656
2400
25666
20800
≥1000
13
0
0
0
0
0
0
0
0
0
0
0
≥1000
0
14
0
0
0
0
0
1152
0
0
1280
0
0
≥1000
0
0
15
0
0
0
0
0
0
0
4
0
420
0
≥1000
0
≥1
0
16
0
0
0
0
0
3328
0
0
656
0
0
≥1000
0
0
≥1
0
17
0
0
0
0
0
0
0
0
0
0
0
≥1000
0
0
0
0
0
18
0
8
0
64
0
9728
0
283584
0
5191380
13824
≥1000
≥1
≥1
≥1
≥1
≥1
≥1
19
0
0
0
0
0
0
0
0
0
0
0
≥1000
0
0
0
0
0
≥1
0
20
0
0
0
0
0
28672
0
0
0
0
0
≥1000
0
0
≥1
0
0
≥1
0
0
21
0
0
0
0
0
0
0
448
0
323680
0
≥1000
0
≥1
0
≥1
0
≥1
0
≥1
0
22
0
0
0
0
0
83968
0
0
0
0
0
≥1000
0
0
≥1
0
0
≥1
0
0
≥1
0
23
0
0
0
0
0
0
0
0
0
0
0
≥1000
0
0
0
0
0
≥1
0
0
0
0
0
24
0
16
0
256
0
245760
147600
21869312
7643200
1.04640703×10¹⁰
796356362
≥1000
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
25
0
0
0
0
0
0
0
0
0
0
0
≥1000
0
0
0
0
0
≥1
0
0
0
0
0
≥1
?
26
0
0
0
0
0
720896
0
0
6909440
0
0
≥1000
0
0
≥1
0
0
≥1
0
0
≥1
0
0
≥1
?
27
0
0
0
0
0
0
0
55232
0
121174272
0
≥1000
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
?
28
0
0
0
0
0
2113536
0
0
6227456
0
0
≥1000
0
0
≥1
0
0
≥1
0
0
≥1
0
0
≥1
?
29
0
0
0
0
0
0
0
0
0
0
0
≥1000
0
0
0
0
0
≥1
0
0
0
0
0
≥1
?
30
0
32
0
1024
0
6193152
0
1.68512753×10¹⁰
2163200
2.10875143×10¹²
2.72048834×10¹⁰
≥1000
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
31
0
0
0
0
0
0
0
0
0
0
0
≥1000
0
0
0
0
0
≥1
0
0
0
0
0
≥1
?
32
0
0
0
0
0
18153472
0
0
6284736
0
0
≥1000
0
0
≥1
0
0
≥1
0
0
≥1
0
0
≥1
?
33
0
0
0
0
0
0
0
5886848
0
3.61400184×10¹¹
0
≥1000
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
?
34
0
0
0
0
0
53215232
0
0
3948480
0
0
≥1000
0
0
≥1
0
0
≥1
0
0
≥1
0
0
≥1
?
35
0
0
0
0
0
0
0
0
0
0
0
≥1
0
0
0
0
0
≥1
0
0
0
0
0
≥1
?
36
0
64
0
4096
0
155975680
70494400
1.29886818×10¹²
2.41921376×10¹¹
4.25091983×10¹⁴
3.60181011×10¹⁴
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
N>0
x
6k
x
6k
x
2k
12k
3k
2k
3k
6k
all
6k
3k
2k
3k
6k
all
6k
3k
2k
3k
6k
all

## Smallest prime reptiles¶

Smallest prime reptile (6Lx6):

polyomino \ n²
10²
11²
12²
L hexomino
1
0
0
0
0
134053888
≥22600000
≥1
≥1
≥1
≥1
≥1

## 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 $A(w; h)$

$G(N(2); x) = \frac{1}{1 - 2x^6} \tag{1}$

$G(N(4); x) = \frac{1}{1 - 4x^6} \tag{2}$

$G(N(6); x) = \frac{1}{1 - 2x^2 - 8x^6} \tag{3}$

$G(N(7); x) = \frac{1 - 144x^{12} - 3648x^{24} + 292864x^{36} - 13500416x^{48} + 1480589312x^{60} + 30601641984x^{72} - 2456721293312x^{84} + 42880953483264x^{96} - 2269391999729664x^{108} - 36028797018963968x^{132}}{1 - 444x^{12} - 18048x^{24} + 747264x^{36} + 15785984x^{48} + 5163712512x^{60} + 178660573184x^{72} - 6268504768512x^{84} - 202791175847936x^{96} - 12055045486936064x^{108} - 226939199972966400x^{120} - 36028797018963968x^{132} - 3602879701896396800x^{144}} \tag{4}$