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.


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 (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 \ h12345678910N>0
10
200
3000
40000
5022P066C0
600002020C0
70000000
800006262C000
9000000000
10044C05252C194194C808808C28582858C1399813998C6615466154C378268378268C
1100000000015503701550370C5k
120000612612C000079876727987672C5k
13000000000≥1≥1C5k
14000019221922C0000≥1≥1C5k
15088C0512512C04104241042C134524134524P47963184796318C≥1≥1C≥1≥1Call
16000060386038C0000≥1≥1C5k
17000000000≥1≥1C5k
1800001898018980C0000≥1≥1C5k
19000000000≥1≥1C5k
2001616C052085208C5964659646C22723542272354C2188209821882098C≥1≥1C≥1≥1C≥1≥1Call
21000000000≥1≥1C5k
220000187442187442C0000≥1≥1C5k
23000000000≥1≥1C5k
240000589076589076C0000≥1≥1C5k
2503232C05338453384C0131052762131052762C1.76809363×10¹⁰1768093636C≥1≥1C≥1≥1C≥1≥1Call
N>0x5kx5k2k5k5k5k5kall

Smallest prime reptiles

Smallest prime reptile (5Px2):

Reptile tilings' solutions count (including symmetric)

polyomino \ n²
P pentomino12P5P3451C5039953P≥1645460432C

Smallest tori tilings

Smallest torus (2x5):

Smallest square torus (5x5):

Smallest odd torus (5x5):

Tori tilings' solutions count (including translations)

w \ h12345678910
100
20000
3000000
400000000
500404000540540440440
6000000005200520000
700000000336033600000
8000000005322053220000000
900000000298802988000000000
1000320320002500025000≥100000≥100000≥100000≥100000≥100000≥100000≥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