# 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.

## L1 octomino¶

Area: 8.

Perimeter: 18.

Size: 2x7.

Is rectangular: no.

Is convex: yes.

Holes: 0.

Order: 2.

Square order: 8.

Prime rectangles: ≥ 4.

## Smallest rectangle tilings¶

Smallest rectangle (2x8):

Smallest square (8x8):

No odd rectangles exist.

## 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
25
26
27
28
29
30
31
32
N>0
1
0
2
0
0
3
0
0
0
4
0
0
0
0
5
0
0
0
0
0
6
0
0
0
0
0
0
7
0
0
0
0
0
0
0
8
0
2
0
4
0
8
0
32
9
0
0
0
0
0
0
0
0
0
10
0
0
0
0
0
0
0
96
0
0
11
0
0
0
0
0
0
0
0
0
0
0
12
0
0
0
0
0
0
0
256
0
0
0
0
13
0
0
0
0
0
0
0
0
0
0
0
0
0
14
0
0
0
0
0
0
0
640
0
0
0
0
0
0
15
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
16
0
4
0
16
0
64
0
1792
972
16776
7776
115776
≥1
≥1
≥1
≥1
17
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
≥1
0
18
0
0
0
0
0
0
0
5120
0
0
0
0
0
0
0
≥1
0
0
19
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
≥1
0
0
0
20
0
0
0
0
0
0
0
14336
0
0
0
0
0
0
0
≥1
0
0
0
≥131072
21
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
≥1
0
0
0
0
0
22
0
0
0
0
0
0
0
38912
0
0
0
0
0
0
0
≥1
0
0
0
0
0
0
23
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
≥1
0
0
0
0
0
0
0
24
0
8
0
64
0
512
0
106496
0
2835808
0
49715712
0
0
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
25
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
≥1
0
0
0
0
0
0
0
0
0
26
0
0
0
0
0
0
0
294912
0
0
0
0
0
0
0
≥1
0
0
0
0
0
0
0
≥1
0
0
27
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
≥1
0
0
0
0
0
0
0
0
0
0
0
28
0
0
0
0
0
0
0
819200
0
0
0
0
0
0
0
≥1
0
0
0
≥1
0
0
0
≥1
0
0
0
≥1
29
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
≥1
0
0
0
0
0
0
0
0
0
0
0
0
0
30
0
0
0
0
0
0
0
2260992
0
0
0
0
0
0
0
≥1
0
0
0
0
0
0
0
≥1
0
0
0
0
0
0
31
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
≥1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
32
0
16
0
256
0
4096
0
6225920
1815696
473812736
88511616
2.12259655×10¹¹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
33
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
≥1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
≥1
?
34
0
0
0
0
0
0
0
17170432
0
0
0
0
0
0
0
≥1
0
0
0
0
0
0
0
≥1
0
0
0
0
0
0
0
≥1
?
35
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
≥1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
≥1
?
36
0
0
0
0
0
0
0
47448064
0
0
0
0
0
0
0
≥1
0
0
0
≥1
0
0
0
≥1
0
0
0
≥1
0
0
0
≥1
?
37
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
≥1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
≥1
?
38
0
0
0
0
0
0
0
131072000
0
0
0
0
0
0
0
≥1
0
0
0
0
0
0
0
≥1
0
0
0
0
0
0
0
≥1
?
39
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
≥1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
≥1
?
40
0
32
0
1024
0
32768
0
361758720
0
7.91663587×10¹¹
0
9.06722280×10¹³
0
0
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
?
41
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
≥1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
≥1
?
42
0
0
0
0
0
0
0
998244352
0
0
0
0
0
0
0
≥1
0
0
0
0
0
0
0
≥1
0
0
0
0
0
0
0
≥1
?
43
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
≥1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
≥1
?
44
0
0
0
0
0
0
0
2.75565772×10¹⁰
0
0
0
33689600
0
0
0
≥1
0
0
0
≥1
0
0
0
≥1
0
0
0
≥1
0
0
0
≥1
?
45
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
≥1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
≥1
?
46
0
0
0
0
0
0
0
7.60846745×10¹⁰
0
0
0
0
0
0
0
≥1
0
0
0
0
0
0
0
≥1
0
0
0
0
0
0
0
≥1
?
47
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
≥1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
≥1
?
48
0
64
0
4096
0
262144
0
2.10050744×10¹¹
3.25038355×10¹⁰
1.32299139×10¹⁴
9.90601519×10¹²
3.87355169×10¹⁶
≥1
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≥1
≥1
≥1
≥1
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≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
N>0
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8k
x
8k
x
8k
x
2k
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8k
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4k
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16k
16k
all
16k
8k
16k
4k
?
?
?
?
?
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?
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?

## Smallest prime reptiles¶

Smallest prime reptile (8L1x4):

polyomino \ n²
L1 octomino
1
0
0
256
0
0
0
≥100000
≥100000

## 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 $A(w; h)$

$N(n; m) = T(n; m) = 0, \qquad 16\nmid nm \tag{1}$

Assume L1 octomino tiles $n\times m$ rectangles for $nm\not\equiv 0\pmod{16}$.

Place numbers in rectangles' cells according to function $F(x,y)\equiv8x+1\pmod{16}$, where $x$ and $y$ are cells' coordinates (zero-based). On the one hand, L1 octomino, no matter how placed, covers sum congruent to $0\pmod{16}$. Then sum covered by all octominoes is also congruent to $0\pmod{16}$. On the other hand, rectangle covers sum congruent to $\sum_{x=0}^{n-1}\sum_{y=0}^{m-1}\left(8x+1\right)$, which is not congruent to $0\pmod{16}$ for $nm\not\equiv 0\pmod{16}$. Contradiction, as octomino tiles this rectangle and thus sum covered by all octominoes should be equal to sum covered by rectangle. Thus only assumption we made is false - L1 octomino doesn't tile $n\times m$ rectangles for $nm\not\equiv 0\pmod{16}$. Q.E.D.