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¶

Area: 4.

Perimeter: 10.

Size: 1x4.

Is rectangular: yes.

Is convex: yes.

Holes: 0.

Order: 1.

Square order: 4.

Odd order: 1.

Prime rectangles: 1.

Smallest rectangle tilings¶

Smallest rectangle and smallest odd rectangle (1x4):

Smallest square (4x4):

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 \ h123456789101112N>0
10
200
3000
411P11C11C22C
500033C0
600044C00
700055C000
811C11C11C77C1515C2525C3737C100100C
90001010C000229229C0
100001414C000454454C00
110001919C000811811C000
1211C11C11C2626C7575C154154C269269C17321732C57645764C1513115131C3434534345C135950135950C
130003636C00037773777C000462186462186C4k
140005050C00078587858C00013562841356284C4k
150006969C0001533915339C00035394333539433C4k
1611C11C11C9595C371371C943943C19491949C3127331273C143765143765C496416496416C14355951435595C1168109111681091Call
17000131131C0006553665536C000≥1≥1C4k
18000181181C000136600136600C000≥1≥1C4k
19000250250C000276535276535C000≥1≥1C4k
2011C11C11C345345C18331833C57735773C1412114121C562728562728C≥1≥1C≥1≥1C≥1≥1C≥1≥1Call
21000476476C00011599421159942C000≥1≥1C4k
22000657657C000≥1≥1C000≥1≥1C4k
23000907907C000≥1≥1C000≥1≥1C4k
2411C11C11C12521252C90579057C3534435344C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1Call
N>04k4k4kall4k4k4kall4k4k4kall

Smallest prime reptiles¶

Smallest prime reptile (4Ix2):

Reptile tilings' solutions count (including symmetric)¶

polyomino \ n²
I tetromino11P1P95C1833P35344C

Smallest tori tilings¶

Smallest torus and smallest odd torus (1x4):

Smallest square torus (4x4):

Tori tilings' solutions count (including translations)¶

w \ h123456789101112
100
20000
3000000
44416166464512512
50000001044104400
6000000419241920000
70000001683216832000000
84416166464678406784012841284522452242159221592179216179216
900000027139627139600000037406837406800
10000000≥500000≥500000000000≥500000≥5000000000
11000000≥500000≥500000000000≥500000≥500000000000
124416166464≥500000≥5000003324332412952129525614456144≥500000≥500000≥500000≥500000≥500000≥500000≥500000≥500000≥500000≥500000

Smallest Baiocchi figures¶

Smallest Baiocchi figure (area 16):

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 $A(w; h)$

$N(1; 4n) = T(1; 4n) = 1, \qquad n \geq 1 \tag{1}$

$N(2; 4n) = T(2; 4n) = 1, \qquad n \geq 1 \tag{2}$

$N(3; 4n) = T(3; 4n) = 1, \qquad n \geq 1 \tag{3}$

"I" tetrominoes can be placed in $1\times 4n$, $2\times 4n$ and $3\times 4n$ rectangles only in one orientation, thus there is maximum one way to place them. One side is divisible by 4, therefore tetrominoes fit perfectly. Q.E.D.

$N(n; m) = T(n; m) = 0, \qquad 4\nmid n,4\nmid m \tag{4}$

Assume I tetromino tiles $n\times m$ rectangles for $4\nmid n,4\nmid m$.

Place numbers in rectangles' cells according to function $F(x,y)\equiv 2x^2+2y^2+12xy+1\pmod{16}$, where $x$ and $y$ are cells' coordinates (zero-based). On the one hand, I tetromino, no matter how placed, covers sum congruent to $0\pmod{16}$. Then sum covered by all tetrominoes is also congruent to $0\pmod{16}$. On the other hand, rectangle covers sum congruent to $\sum_{x=0}^{n-1}\sum_{y=0}^{m-1}\left(2x^2+2y^2+12xy+1\right)$, which is not congruent to $0\pmod{16}$ for $4\nmid n,4\nmid m$. Contradiction, as tetromino tiles this rectangle and thus sum covered by all tetrominoes should be equal to sum covered by rectangle. Thus only assumption we made is false - I tetromino doesn't tile $n\times m$ rectangles for $4\nmid n,4\nmid m$. Q.E.D.

$T(4n; m) = 1 \tag{5}$

$G(T; x; y) = \frac{xy\left(x^3 y^3+x^3 y^2+x^3 y+x^3+x^2 y^3+x y^3+y^3\right)}{\left(1-x^4\right)\left(1-y^4\right)} \tag{6}$