POLYOMINO TILINGS

Polyomino Tilings

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You may also see list of all polyomino sets for which data is available here.


P pentomino

Area: 5.

Perimeter: 10.

Size: 2x3.

Is rectangular: no.

Is convex: yes.

Holes: 0.

Order: 2.

Square order: 20.

Odd order: 21.

Prime rectangles: 2.

Smallest rectangle tilings

Smallest rectangle (2x5):

Smallest square (10x10):

Smallest odd rectangle (7x15):

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
2
0
6
0
6
0
0
0
0
20
0
7
0
0
0
0
0
0
0
8
0
0
0
0
62
0
0
0
9
0
0
0
0
0
0
0
0
0
10
0
4
0
52
194
808
2858
13998
66154
378268
11
0
0
0
0
0
0
0
0
0
1550370
0
12
0
0
0
0
612
0
0
0
0
7987672
0
0
13
0
0
0
0
0
0
0
0
0
≥1
0
0
0
14
0
0
0
0
1922
0
0
0
0
≥1
0
0
0
0
15
0
8
0
512
0
41042
134524
4796318
≥1
≥1
≥1
≥1
≥1
≥1
≥1
16
0
0
0
0
6038
0
0
0
0
≥1
0
0
0
0
≥1
0
17
0
0
0
0
0
0
0
0
0
≥1
0
0
0
0
≥1
0
0
18
0
0
0
0
18980
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
19
0
0
0
0
0
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
20
0
16
0
5208
59646
2272354
21882098
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
21
0
0
0
0
0
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
≥1
5k
22
0
0
0
0
187442
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
≥1
5k
23
0
0
0
0
0
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
≥1
5k
24
0
0
0
0
589076
0
0
0
0
≥1
0
0
0
0
≥1
0
0
0
0
≥1
5k
25
0
32
0
53384
0
131052762
1.76809363×10¹⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
all
N>0
x
5k
x
5k
2k
5k
5k
5k
5k
all
5k
5k
5k
5k
all
5k
5k
5k
5k
all

Smallest prime reptiles

Smallest prime reptile (5Px2):

Reptile tilings' solutions count (including symmetric)

polyomino \ n²
P pentomino
1
2
5
3451
5039953
≥1645460432

Smallest tori tilings

Smallest torus (2x5):

Smallest square torus (5x5):

Smallest odd torus (5x5):

Tori tilings' solutions count (including translations)

w \ h
1
2
3
4
5
6
7
8
9
10
1
0
2
0
0
3
0
0
0
4
0
0
0
0
5
0
40
0
540
440
6
0
0
0
0
5200
0
7
0
0
0
0
3360
0
0
8
0
0
0
0
53220
0
0
0
9
0
0
0
0
29880
0
0
0
0
10
0
320
0
25000
≥100000
≥100000
≥100000
≥100000
≥100000
≥100000

Smallest Baiocchi figures

Smallest Baiocchi figure (area 20):

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 $A(w; h)$

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

"P" pentomino has minimal dimension of $2$ thus it does not fit into $1 \times n$ rectangle. Q.E.D.

$N(2; n) = 2 \times N(2; n - 5), \qquad n \geq 6 \tag{2}$

Top left corner (marked with dot on image) may be covered in only three ways. Only two of them are valid, because third leave untileable hole (marked with cross on image). Then, placing next pentomino in first two ways a $2 \times 5$ rectangle is covered. Remaining part is $2 \times (n-5)$ rectangle. So whole rectangle can be tiled in $2 \times N(2; n - 5)$ ways. Q.E.D.

$N(3; n) = T(3; n) = 0, \qquad n \geq 1 \tag{3}$

Top left corner (marked with dot on image) may be covered in only six ways. Only two of them are valid, because remaining ways leave untileable holes (marked with crosses on image). However, placing pentomino in first two ways is possible only in one way and then it also leaves untileable holes. Q.E.D.

$N(4; n) = 15 \times N(4; n - 5) - 54 \times N(4; n - 10) + 56 \times N(4; n - 15), \qquad n \geq 16 \tag{5}$

$N(5; n) = 2 \times N(2; n - 2) + 2 \times N(2; n - 4) + 5 \times N(2; n - 6), \qquad n \geq 7 \tag{5}$

$N(5; 2n + 1) = T(5; 2n + 1) = 0, \qquad n \geq 0 \tag{6}$

See Also

N pentominoR pentomino