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.


L tetromino and X pentomino

Prime rectangles: 23.

Smallest rectangle tilings

Smallest rectangle (5x6):

Smallest square (6x6):

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
1-4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
N>0
1-4
0
5
0
0
6
0
4
2
7
0
0
0
0
8
0
0
0
0
0
9
0
0
424
0
0
0
10
0
32
208
14288
136
230672
1356846
11
0
0
24
0
466
0
124653460
0
12
0
16
4
3240
6464
2773498
19239156
≥1
≥1
13
0
0
55848
0
25124
0
5.48436858×10¹⁰
0
≥1
0
14
0
3720
27456
1555060
226816
4.68291631×10¹⁰
1.39450791×10¹¹
≥1
≥1
≥1
≥1
15
0
0
6728
0
1338150
0
1.36255894×10¹³
0
≥1
0
≥1
0
16
0
272
760
1439672
10898896
7.19972252×10¹⁰
3.73284706×10¹²
≥1
≥1
≥1
≥1
≥1
≥1
17
0
0
7621960
0
63050364
0
7.66591455×10¹⁴
0
≥1
0
≥1
0
≥1
0
18
0
63216
3795240
1.68735222×10¹⁰
440659642
7.38892669×10¹³
1.62216498×10¹⁵
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
19
0
0
1596306
0
2.71679907×10¹⁰
0
1.56444744×10¹⁷
0
≥1
0
≥1
0
≥1
0
≥1
0
20
0
25232
154624
1.06613176×10¹⁰
1.84592759×10¹¹
5.28727844×10¹⁴
5.66929416×10¹⁶
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
21
0
0
1.06080419×10¹⁰
0
1.14106220×10¹²
0
9.35868375×10¹⁸
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
?
22
0
4616830
534480960
2.54588019×10¹²
7.45836015×10¹²
5.15876043×10¹⁷
1.93058807×10¹⁹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
23
0
0
324752424
0
4.64695893×10¹³
0
≥1.84467440×10²⁰
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
?
24
0
662892
31636448
3.24455583×10¹²
2.98164536×10¹⁴
1.34690934×10¹⁸
≥1.84467440×10²⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
25
0
0
1.49186594×10¹²
0
1.85572528×10¹⁵
0
≥1.84467440×10²⁰
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
?
26
0
94265682
7.59583244×10¹¹
2.00365557×10¹⁵
1.17386161×10¹⁶
≥1.84467440×10²⁰
≥1.84467440×10²⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
27
0
0
5.97155308×10¹¹
0
7.29281107×10¹⁶
0
≥1.84467440×10²⁰
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
?
28
0
40795284
6.19160631×10¹⁰
1.85904557×10¹⁵
4.56802192×10¹⁷
≥1.84467440×10²⁰
≥1.84467440×10²⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
29
0
0
2.10615264×10¹⁴
0
2.82895010×10¹⁸
0
≥1.84467440×10²⁰
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
?
30
0
5.44345964×10¹⁰
1.08281093×10¹⁴
3.37935387×10¹⁷
1.75895853×10¹⁹
≥1.84467440×10²⁰
≥1.84467440×10²⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
31
0
0
1.02664211×10¹⁴
0
1.08565290×10²⁰
0
≥1.84467440×10²⁰
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
?
32
0
1.17568696×10¹⁰
1.15080283×10¹³
5.40077804×10¹⁷
≥1.84467440×10²⁰
≥1.84467440×10²⁰
≥1.84467440×10²⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
33
0
0
2.97399547×10¹⁶
0
≥1.84467440×10²⁰
0
≥1.84467440×10²⁰
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
?
34
0
1.24466730×10¹²
1.54329691×10¹⁶
≥1.84467440×10²⁰
≥1.84467440×10²⁰
≥1.84467440×10²⁰
≥1.84467440×10²⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
35
0
0
1.68486749×10¹⁶
0
≥1.84467440×10²⁰
0
≥1.84467440×10²⁰
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
?
36
0
5.99170691×10¹¹
2.04400292×10¹⁵
≥1.84467440×10²⁰
≥1.84467440×10²⁰
≥1.84467440×10²⁰
≥1.84467440×10²⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
37
0
0
4.19297473×10¹⁸
0
≥1.84467440×10²⁰
0
≥1.84467440×10²⁰
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
?
38
0
6.25011479×10¹³
2.19577436×10¹⁸
≥1.84467440×10²⁰
≥1.84467440×10²⁰
≥1.84467440×10²⁰
≥1.84467440×10²⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
39
0
0
2.67449783×10¹⁸
0
≥1.84467440×10²⁰
0
≥1.84467440×10²⁰
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
?
40
0
1.80233881×10¹³
3.49880552×10¹⁷
≥1.84467440×10²⁰
≥1.84467440×10²⁰
≥1.84467440×10²⁰
≥1.84467440×10²⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
41
0
0
≥1.84467440×10²⁰
0
≥1.84467440×10²⁰
0
≥1.84467440×10²⁰
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
?
42
0
1.53897840×10¹⁵
≥1.84467440×10²⁰
≥1.84467440×10²⁰
≥1.84467440×10²⁰
≥1.84467440×10²⁰
≥1.84467440×10²⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
43
0
0
≥1.84467440×10²⁰
0
≥1.84467440×10²⁰
0
≥1.84467440×10²⁰
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
?
44
0
8.25677114×10¹⁴
5.81504621×10¹⁹
≥1.84467440×10²⁰
≥1.84467440×10²⁰
≥1.84467440×10²⁰
≥1.84467440×10²⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
45
0
0
≥1.84467440×10²⁰
0
≥1.84467440×10²⁰
0
≥1.84467440×10²⁰
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
?
46
0
7.03630106×10¹⁶
≥1.84467440×10²⁰
≥1.84467440×10²⁰
≥1.84467440×10²⁰
≥1.84467440×10²⁰
≥1.84467440×10²⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
47
0
0
≥1.84467440×10²⁰
0
≥1.84467440×10²⁰
0
≥1.84467440×10²⁰
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
?
48
0
2.53837988×10¹⁶
≥1.84467440×10²⁰
≥1.84467440×10²⁰
≥1.84467440×10²⁰
≥1.84467440×10²⁰
≥1.84467440×10²⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
49
0
0
≥1.84467440×10²⁰
0
≥1.84467440×10²⁰
0
≥1.84467440×10²⁰
0
≥1
0
≥1
0
≥1
0
≥1
0
≥1
?
50
0
1.82585880×10¹⁸
≥1.84467440×10²⁰
≥1.84467440×10²⁰
≥1.84467440×10²⁰
≥1.84467440×10²⁰
≥1.84467440×10²⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
N>0
x
2k
all
2k
all
2k
all
2k
all
2k
all
2k
all
2k
all
2k
all

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 $A(w; h)$

$N(2n + 1; 2m + 1) = T(2n + 1; 2m + 1) = 0 \tag{1}$

Assume L tetromino and X pentomino tile $(2n + 1)\times(2m + 1)$ rectangle. Place numbers in rectangle's cells according to function $F(x,y)\equiv(-1)^{x+y}\pmod{3}$, where $x$ and $y$ are cells' coordinates (zero-based). On the one hand, L tetromino and X pentomino, no matter how placed, cover sum congruent to $0\pmod{3}$. Then sum covered by all polyominoes is also congruent to $0\pmod{3}$. On the other hand, rectangle covers sum congruent to $\sum_{x=0}^{(2n+1)-1}\sum_{y=0}^{(2m+1)-1}(-1)^{x+y}\equiv1\pmod{3}$. Contradiction, as L tetromino and X pentomino tile this rectangle and thus sum covered by all polyominoes should be equal to sum covered by rectangle. Thus only assumption we made is false - L tetromino and X pentomino don't tile $(2n + 1)\times(2m + 1)$ rectangle. Q.E.D.

See Also

L tetromino and W pentominoL tetromino and Y pentomino