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.


I tetromino and T1 hexomino

Prime rectangles: ≥ 5.

Smallest rectangle tilings

Smallest rectangle (4x5):

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
1-3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
1-3
0
4
0
0
5
0
2
0
6
0
4
0
0
7
0
6
0
0
0
8
0
8
46
100
180
602
9
0
16
0
0
0
2988
?
10
0
32
0
0
0
≥1
?
?
11
0
56
0
0
0
≥1
?
?
?
12
0
88
586
1632
4060
≥1
≥1
≥1
≥1
≥1
13
0
140
0
0
0
≥1
?
?
?
≥1
?
14
0
236
0
0
0
≥1
?
?
?
≥1
?
?
15
0
402
0
0
0
≥1
?
?
?
≥1
?
?
?
16
0
664
6634
24496
88552
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
17
0
≥1
?
?
?
≥1
?
?
?
≥1
?
?
?
≥1
?
18
0
≥1
?
?
?
≥1
?
?
?
≥1
?
?
?
≥1
?
?
19
0
≥1
?
?
?
≥1
?
?
?
≥1
?
?
?
≥1
?
?
?
20
0
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
N>0
x
all
?
?
?
all
?
?
?
all
?
?
?
all
?
?
?
?

Smallest common multiples

Smallest common multiple (area 24):

Smallest known common multiple without holes (area 36):

Smallest known convex common multiple (area 48):

Common multiples' solutions count (excluding symmetric)

area
12
24
36
48
solutions
?
≥2
≥1
≥1

Attributions

  1. Smallest rectangle and square found by Dmitry Grekov
  2. Solutions counted by Dmitry Grekov
  3. Smallest common multiple found by Jorge Mireles
  4. Holeless common multiple found by Dmitry Grekov
  5. Convex common multiple found by Dmitry Grekov

See Also

I tetromino and G hexominoI tetromino and X2 hexomino