# 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.

Odd order: ∞.

Prime rectangles: 4.

## Smallest rectangle tilings¶

Smallest rectangle (2x8):

Smallest square (8x8):

No odd rectangles exist.

## 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 \ h1234567891011121314151617181920N>0
10
200
3000
40000
500000
6000000
70000000
8022P044C088C03232C
9000000000
1000000009696C00
1100000000000
120000000256256C00000
130000000000000
140000000640640C000000
15000000000000000
16044C01616C06464C017921792C972972P1677616776C77767776C115776115776C≥1≥1C≥1≥1C≥1≥1C≥1≥1C
17000000000000000≥1≥1C0
18000000051205120C0000000≥1≥1C00
19000000000000000≥1≥1C000
2000000001433614336C00000000≥1≥1C000≥131072≥131072P
21000000000000000≥1≥1C0000?
2200000003891238912C0000000≥1≥1C0000?
23000000000000000≥1≥1C0000?
24088C06464C0512512C0106496106496C028358082835808C04971571249715712C000≥1≥1C0≥1≥1C0≥1≥1C?
25000000000000000≥1≥1C0000?
260000000294912294912C0000000≥1≥1C0000?
27000000000000000≥1≥1C0000?
280000000819200819200C00000000≥1≥1C000≥1≥1C?
29000000000000000≥1≥1C0000?
30000000022609922260992C0000000≥1≥1C0000?
31000000000000000≥1≥1C0000?
3201616C0256256C040964096C062259206225920C18156961815696C473812736473812736C8851161688511616C2.12259655×10¹¹21225965568C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C?
33000000000000000≥1≥1C0000?
3400000001717043217170432C0000000≥1≥1C0000?
35000000000000000≥1≥1C0000?
3600000004744806447448064C00000000≥1≥1C000≥1≥1C?
37000000000000000≥1≥1C0000?
380000000131072000131072000C0000000≥1≥1C0000?
39000000000000000≥1≥1C0000?
4003232C010241024C03276832768C0361758720361758720C07.91663587×10¹¹79166358784C09.06722280×10¹³9067222806528C000≥1≥1C0≥1≥1C0≥1≥1C?
41000000000000000≥1≥1C0000?
420000000998244352998244352C0000000≥1≥1C0000?
43000000000000000≥1≥1C0000?
4400000002.75565772×10¹⁰2755657728C0003368960033689600P000≥1≥1C000≥1≥1C?
45000000000000000≥1≥1C0000?
4600000007.60846745×10¹⁰7608467456C0000000≥1≥1C0000?
47000000000000000≥1≥1C0000?
4806464C040964096C0262144262144C02.10050744×10¹¹21005074432C3.25038355×10¹⁰3250383552C1.32299139×10¹⁴13229913969152C9.90601519×10¹²990601519104C3.87355169×10¹⁶3873551696052224C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C?
N>0x8kx8kx8kx2k16k8k16k4k16k16k16kall16k8k16k4k

## Smallest prime reptiles¶

Smallest prime reptile (8L1x4):

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

polyomino \ n²
L1 octomino100256P000≥100000P≥100000P

## 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.