POLYOMINO TILINGS

Polyomino Tilings

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D hexomino

Area: 6.

Size: 2x4.

Holes: 0.

Order: 4.

Square order: 24.

Odd order: ∞.

Prime rectangles: 2.

Smallest rectangle tilings

Smallest rectangle (4x6):

Smallest square (12x12):

No odd rectangles exist.

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
1-3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
N>0
1-3
0
4
0
0
5
0
0
0
6
0
1
0
0
7
0
0
0
0
0
8
0
0
0
2
0
0
9
0
0
0
0
0
0
0
10
0
0
0
0
0
0
0
0
11
0
0
0
0
0
0
0
0
0
12
0
2
2
4
0
16
20
30
72
256
13
0
0
0
0
0
0
0
0
0
192
0
14
0
0
0
0
0
0
0
0
0
358
0
0
15
0
0
0
0
0
0
0
0
0
842
0
0
0
16
0
0
0
8
0
0
0
0
0
2284
0
0
0
0
17
0
0
0
0
0
0
0
0
0
4352
0
0
0
0
0
18
0
4
0
0
0
128
0
0
0
7968
0
0
0
≥1
0
0
19
0
0
0
0
0
0
0
0
0
≥1
0
0
0
0
0
0
0
20
0
0
0
16
0
0
0
0
0
≥1
0
0
0
0
0
≥1
0
0
21
0
0
0
0
0
0
0
0
0
≥1
0
0
0
0
0
0
0
0
12k
22
0
0
0
0
0
0
0
0
0
≥1
0
0
0
0
0
0
0
0
12k
23
0
0
0
0
0
0
0
0
0
≥1
0
0
0
0
0
0
0
0
12k
24
0
8
6
32
0
1024
960
2058
12544
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
all
25
0
0
0
0
0
0
0
0
0
≥1
0
0
0
0
0
0
0
0
12k
26
0
0
0
0
0
0
0
0
0
≥1
0
0
0
0
0
0
0
0
12k
27
0
0
0
0
0
0
0
0
0
≥1
0
0
0
0
0
0
0
0
12k
28
0
0
0
64
0
0
0
0
0
≥1
0
0
0
0
0
≥1
0
0
6k
29
0
0
0
0
0
0
0
0
0
≥1
0
0
0
0
0
0
0
0
12k
30
0
16
0
0
0
8192
0
0
0
≥1
0
0
0
≥1
0
0
0
≥1
4k
N>0
x
6k
12k
4k
x
6k
12k
12k
12k
all
12k
12k
12k
6k
12k
4k
12k
6k

Smallest prime reptiles

Smallest prime reptile (6Dx6):

Reptile tilings' solutions count (including symmetric)

polyomino \ n²
10²
11²
12²
13²
D hexomino
1
0
0
0
0
≥1
0
0
0
0
≥1
≥1
≥1

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 $A(w; h)$

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

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

$N(3; n) = T(3; n) = 0 \tag{3}$

$N(7; n) = T(7; n) = 0 \tag{4}$

$N(n; m) = T(n; m) = 0, \qquad 6\nmid n,6\nmid m \tag{5}$

Assume D hexomino tiles $n\times m$ rectangles for $6\nmid n,6\nmid m$.

Place numbers in rectangles' cells according to function $F(x,y)\equiv (-1)^{\left\lfloor\frac{y}{3}\right\rfloor+x}+(-1)^{\left\lfloor\frac{x}{3}\right\rfloor+y}+2-2(-1)^{x+y}\pmod{12}$, where $x$ and $y$ are cells' coordinates (zero-based). On the one hand, D hexomino, no matter how placed, covers sum congruent to $0\pmod{12}$. Then sum covered by all hexominoes is also congruent to $0\pmod{12}$. On the other hand, rectangle covers sum congruent to $\sum_{x=0}^{n-1}\sum_{y=0}^{m-1}F(x,y)$, which is not congruent to $0\pmod{12}$ for $6\nmid n,6\nmid m$. Contradiction, as hexomino tiles this rectangle and thus sum covered by all hexominoes should be equal to sum covered by rectangle. Thus only assumption we made is false - D hexomino doesn't tile $n\times m$ rectangles for $6\nmid n,6\nmid m$. Q.E.D.

$N(n; m) = T(n; m) = 0, \qquad 4\nmid n,4\nmid m \tag{6}$

Proved in [1].

Attributions

  1. Michael Reid, Klarner Systems and Tiling Boxes with Polyominoes, Journal of Combinatorial Theory, Series A 111 (2005), no. 1, pp. 89-105. (http://www.cflmath.com/~reid/Research/Ksystem/index.html)

See Also

C hexominoF hexomino