Loading [MathJax]/extensions/TeX/mathchoice.js

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 - 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
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
1
1
1
1
1
1
1
2
9
0
0
0
0
0
0
0
3
0
10
0
0
0
0
0
0
0
4
0
0
11
0
0
0
0
0
0
0
5
0
0
0
12
0
0
0
0
0
0
0
6
0
0
0
0
13
0
0
0
0
0
0
0
7
0
0
0
0
0
14
0
0
0
0
0
0
0
8
0
0
0
0
0
0
15
0
0
0
0
0
0
0
9
0
0
0
0
0
0
0
16
1
1
1
1
1
1
1
11
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
17
0
0
0
0
0
0
0
≥1
0
0
0
0
0
0
0
≥1
0
18
0
0
0
0
0
0
0
≥1
0
0
0
0
0
0
0
≥1
0
0
19
0
0
0
0
0
0
0
≥1
0
0
0
0
0
0
0
≥1
0
0
0
20
0
0
0
0
0
0
0
≥1
0
0
0
0
0
0
0
≥1
0
?
0
?
21
0
0
0
0
0
0
0
≥1
0
0
0
0
0
0
0
≥1
0
0
0
0
?
N>0
8k
8k
8k
8k
8k
8k
8k
all
8k
8k
8k
8k
8k
8k
8k
all
?
?
?
?

Smallest prime reptiles

Smallest prime reptile (8I1x2):

Reptile tilings' solutions count (including symmetric)

polyomino \ n²
I1 octomino
1
1
1
1
1
1

Formulas

N(w;h) - number of ways to tile w×h rectangle (including symmetric solutions)

T(w;h)={1,N(w;h)10,else - tileability function, 1 if tiles rectangle, 0 otherwise

A(w;h)=(N(w;h))1wh - average number of ways to tile cell in w×h rectangle (including symmetric solutions)

G(T;x;y)=w=1h=1T(w;h)xwyh - bivariate generating function of T(w;h)

G(A;x;y)=w=1h=1A(w;h)xwyh - bivariate generating function of A(w;h)

N(n;m)=T(n;m)=0,8n,8m

Assume I1 octomino tiles n×m rectangles for 8n,8m.

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