POLYOMINO TILINGS

Polyomino Tilings

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


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 (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 \ h1-3456789101112N>0
1-30
400
5000
6011P00
700000
800022C00
90000000
1000000000
11000000000
12022C22P44C01616C2020C3030C7272C256256C
13000000000192192C12k
14000000000358358C12k
15000000000842842C12k
1600088C0000022842284C6k
1700000000043524352C12k
18044C000128128C00079687968C4k
19000000000≥1≥1C12k
200001616C00000≥1≥1C6k
21000000000≥1≥1C12k
22000000000≥1≥1C12k
23000000000≥1≥1C12k
24088C66C3232C010241024C960960C20582058C1254412544C≥1≥1Call
25000000000≥1≥1C12k
26000000000≥1≥1C12k
27000000000≥1≥1C12k
280006464C00000≥1≥1C6k
29000000000≥1≥1C12k
3001616C00081928192C000≥1≥1C4k
N>0x6k12k4kx6k12k12k12kall

Smallest prime reptiles

Smallest prime reptile (6Dx6):

Reptile tilings' solutions count (including symmetric)

polyomino \ n²10²11²12²13²
D hexomino10000≥1P0000≥1P≥1P≥1P

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