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.


I1 octomino

Area: 8.

Perimeter: 18.

Size: 1x8.

Is rectangular: yes.

Is convex: yes.

Holes: 0.

Order: 1.

Square order: 8.

Odd order: 1.

Prime rectangles: 1.

Smallest rectangle tilings

Smallest rectangle and smallest odd rectangle (1x8):

Smallest square (8x8):

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 \ h12345678N>0
10
200
3000
40000
500000
6000000
70000000
811P11C11C11C11C11C11C22C
9000000033C8k
10000000044C8k
11000000055C8k
12000000066C8k
13000000077C8k
14000000088C8k
15000000099C8k
1611C11C11C11C11C11C11C1111Call
N>08k8k8k8k8k8k8kall

Smallest prime reptiles

Smallest prime reptile (8I1x2):

Reptile tilings' solutions count (including symmetric)

polyomino \ n²
I1 octomino11P1P1C1P1C

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 $A(w; h)$

$N(n; m) = T(n; m) = 0, \qquad 8\nmid n,8\nmid m \tag{1}$

Assume I1 octomino tiles $n\times m$ rectangles for $8\nmid n,8\nmid m$.

Place numbers in rectangles' cells according to function $F(x,y)\equiv 8+(-1)^{\left\lfloor\frac{x}{4}\right\rfloor + \left\lfloor\frac{y}{4}\right\rfloor} + (-1)^{\left\lfloor-\frac{x}{4}\right\rfloor + \left\lfloor-\frac{y}{4}\right\rfloor} + (-1)^{\left\lfloor\frac{x}{2}\right\rfloor + \left\lfloor\frac{y}{2}\right\rfloor} + (-1)^{\left\lfloor-\frac{x}{2}\right\rfloor + \left\lfloor-\frac{y}{2}\right\rfloor}\pmod{16}$, where $x$ and $y$ are cells' coordinates (zero-based). On the one hand, I 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 $\sum_{x=0}^{n-1}\sum_{y=0}^{m-1}\left(8+(-1)^{\left\lfloor\frac{x}{4}\right\rfloor + \left\lfloor\frac{y}{4}\right\rfloor} + (-1)^{\left\lfloor-\frac{x}{4}\right\rfloor + \left\lfloor-\frac{y}{4}\right\rfloor} + (-1)^{\left\lfloor\frac{x}{2}\right\rfloor + \left\lfloor\frac{y}{2}\right\rfloor} + (-1)^{\left\lfloor-\frac{x}{2}\right\rfloor + \left\lfloor-\frac{y}{2}\right\rfloor}\right)$, which is not congruent to $0\pmod{16}$ for $8\nmid n,8\nmid m$. 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 - I octomino doesn't tile $n\times m$ rectangles for $8\nmid n,8\nmid m$. Q.E.D.

See Also

G octominoI2 octomino