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Area: 6.

Size: 2x4.

Holes: 0.

Order: 4.

Square order: 24.

Prime rectangles: ≥ 2.

Smallest rectangle (4x6):

Smallest square (12x12):

No odd rectangles exist.

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

0

0

7

0

0

0

0

0

8

0

0

0

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

0

13

0

0

0

0

0

0

0

0

0

0

14

0

0

0

0

0

0

0

0

0

0

0

15

0

0

0

0

0

0

0

0

0

0

0

0

16

0

0

0

0

0

0

0

0

0

0

0

0

17

0

0

0

0

0

0

0

0

0

0

0

0

0

0

18

0

0

0

0

0

0

0

0

0

0

0

0

19

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

20

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

21

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

12k

22

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

12k

23

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

12k

24

0

0

all

25

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

12k

26

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

12k

27

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

12k

28

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

6k

29

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

12k

30

0

0

0

0

0

0

0

0

0

0

0

0

0

4k

N>0

x

6k

12k

4k

x

6k

12k

12k

12k

all

12k

12k

12k

6k

12k

4k

12k

6k

Smallest prime reptile (6Dx6):

$N(w; h)$ - 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].

- 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)