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Area: 6.

Size: 2x4.

Holes: 0.

Order: 4.

Square order: 24.

Odd order: ∞.

Prime rectangles: 2.

Smallest rectangle (4x6):

Smallest square (12x12):

No odd rectangles exist.

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 \ h | 1-3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | N>0 |
---|---|---|---|---|---|---|---|---|---|---|---|

1-3 | 0 | ||||||||||

4 | 0 | 0 | |||||||||

5 | 0 | 0 | 0 | ||||||||

6 | 0 | 11P | 0 | 0 | |||||||

7 | 0 | 0 | 0 | 0 | 0 | ||||||

8 | 0 | 0 | 0 | 22C | 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 | 22C | 22P | 44C | 0 | 1616C | 2020C | 3030C | 7272C | 256256C | |

13 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 192192C | 12k |

14 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 358358C | 12k |

15 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 842842C | 12k |

16 | 0 | 0 | 0 | 88C | 0 | 0 | 0 | 0 | 0 | 22842284C | 6k |

17 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 43524352C | 12k |

18 | 0 | 44C | 0 | 0 | 0 | 128128C | 0 | 0 | 0 | 79687968C | 4k |

19 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ≥1≥1C | 12k |

20 | 0 | 0 | 0 | 1616C | 0 | 0 | 0 | 0 | 0 | ≥1≥1C | 6k |

21 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ≥1≥1C | 12k |

22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ≥1≥1C | 12k |

23 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ≥1≥1C | 12k |

24 | 0 | 88C | 66C | 3232C | 0 | 10241024C | 960960C | 20582058C | 1254412544C | ≥1≥1C | all |

25 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ≥1≥1C | 12k |

26 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ≥1≥1C | 12k |

27 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ≥1≥1C | 12k |

28 | 0 | 0 | 0 | 6464C | 0 | 0 | 0 | 0 | 0 | ≥1≥1C | 6k |

29 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ≥1≥1C | 12k |

30 | 0 | 1616C | 0 | 0 | 0 | 81928192C | 0 | 0 | 0 | ≥1≥1C | 4k |

N>0 | x | 6k | 12k | 4k | x | 6k | 12k | 12k | 12k | all |

Smallest prime reptile (6Dx6):

polyomino \ n² | 1² | 2² | 3² | 4² | 5² | 6² | 7² | 8² | 9² | 10² | 11² | 12² | 13² |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

D hexomino | 1 | 0 | 0 | 0 | 0 | ≥1P | 0 | 0 | 0 | 0 | ≥1P | ≥1P | ≥1P |

$N(w; h)$ - 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)