POLYOMINO TILINGS

Polyomino Tilings

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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 - 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
1
1
1
2
5
0
0
0
3
0
6
0
0
0
4
0
0
7
0
0
0
5
0
0
0
8
1
1
1
7
15
25
37
100
9
0
0
0
10
0
0
0
229
0
10
0
0
0
14
0
0
0
454
0
0
11
0
0
0
19
0
0
0
811
0
0
0
12
1
1
1
26
75
154
269
1732
5764
15131
34345
135950
13
0
0
0
36
0
0
0
3777
0
0
0
462186
0
14
0
0
0
50
0
0
0
7858
0
0
0
1356284
0
0
15
0
0
0
69
0
0
0
15339
0
0
0
3539433
0
0
0
16
1
1
1
95
371
943
1949
31273
143765
496416
1435595
11681091
≥1
≥1
≥1
≥1
17
0
0
0
131
0
0
0
65536
0
0
0
≥1
0
0
0
≥1
0
18
0
0
0
181
0
0
0
136600
0
0
0
≥1
0
0
0
≥1
0
0
19
0
0
0
250
0
0
0
276535
0
0
0
≥1
0
0
0
≥1
0
0
0
20
1
1
1
345
1833
5773
14121
562728
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
21
0
0
0
476
0
0
0
1159942
0
0
0
≥1
0
0
0
≥1
0
0
0
≥1
4k
22
0
0
0
657
0
0
0
≥1
0
0
0
≥1
0
0
0
≥1
0
0
0
≥1
4k
23
0
0
0
907
0
0
0
≥1
0
0
0
≥1
0
0
0
≥1
0
0
0
≥1
4k
24
1
1
1
1252
9057
35344
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
all
N>0
4k
4k
4k
all
4k
4k
4k
all
4k
4k
4k
all
4k
4k
4k
all
4k
4k
4k
all

Smallest prime reptiles

Smallest prime reptile (4Ix2):

Reptile tilings' solutions count (including symmetric)

polyomino \ n²
I tetromino
1
1
1
95
1833
35344

Smallest tori tilings

Smallest torus and smallest odd torus (1x4):

Smallest square torus (4x4):

Tori tilings' solutions count (including translations)

w \ h
1
2
3
4
5
6
7
8
9
10
11
12
1
0
2
0
0
3
0
0
0
4
4
16
64
512
5
0
0
0
1044
0
6
0
0
0
4192
0
0
7
0
0
0
16832
0
0
0
8
4
16
64
67840
1284
5224
21592
179216
9
0
0
0
271396
0
0
0
374068
0
10
0
0
0
≥500000
0
0
0
≥500000
0
0
11
0
0
0
≥500000
0
0
0
≥500000
0
0
0
12
4
16
64
≥500000
3324
12952
56144
≥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.

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

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

See Also

TetrominoesL tetromino