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Area: 8.

Perimeter: 18.

Size: 1x8.

Is rectangular: yes.

Is convex: yes.

Holes: 0.

Order: 1.

Square order: 8.

Odd order: 1.

Prime rectangles: 1.

Smallest rectangle and smallest odd rectangle (1x8):

Smallest square (8x8):

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 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | N>0 |
---|---|---|---|---|---|---|---|---|---|

1 | 0 | ||||||||

2 | 0 | 0 | |||||||

3 | 0 | 0 | 0 | ||||||

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

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

6 | 0 | 0 | 0 | 0 | 0 | 0 | |||

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

8 | 11P | 11C | 11C | 11C | 11C | 11C | 11C | 22C | |

9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 33C | 8k |

10 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 44C | 8k |

11 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 55C | 8k |

12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 66C | 8k |

13 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 77C | 8k |

14 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 88C | 8k |

15 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 99C | 8k |

16 | 11C | 11C | 11C | 11C | 11C | 11C | 11C | 1111C | all |

N>0 | 8k | 8k | 8k | 8k | 8k | 8k | 8k | all |

Smallest prime reptile (8I1x2):

polyomino \ n² | 1² | 2² | 3² | 4² | 5² | 6² |
---|---|---|---|---|---|---|

I1 octomino | 1 | 1P | 1P | 1C | 1P | 1C |

$N(w; h)$ - 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 $A(w; h)$

$N(n; m) = T(n; m) = 0, \qquad 8\nmid n,8\nmid m \tag{1}$

Assume I1 octomino tiles $n\times m$ rectangles for $8\nmid n,8\nmid m$.

Place numbers in rectangles' cells according to function $F(x,y)\equiv 8+(-1)^{\left\lfloor\frac{x}{4}\right\rfloor + \left\lfloor\frac{y}{4}\right\rfloor} + (-1)^{\left\lfloor-\frac{x}{4}\right\rfloor + \left\lfloor-\frac{y}{4}\right\rfloor} + (-1)^{\left\lfloor\frac{x}{2}\right\rfloor + \left\lfloor\frac{y}{2}\right\rfloor} + (-1)^{\left\lfloor-\frac{x}{2}\right\rfloor + \left\lfloor-\frac{y}{2}\right\rfloor}\pmod{16}$, where $x$ and $y$ are cells' coordinates (zero-based). On the one hand, I octomino, no matter how placed, covers sum congruent to $0\pmod{16}$. Then sum covered by all octominoes is also congruent to $0\pmod{16}$. On the other hand, rectangle covers sum congruent to $\sum_{x=0}^{n-1}\sum_{y=0}^{m-1}\left(8+(-1)^{\left\lfloor\frac{x}{4}\right\rfloor + \left\lfloor\frac{y}{4}\right\rfloor} + (-1)^{\left\lfloor-\frac{x}{4}\right\rfloor + \left\lfloor-\frac{y}{4}\right\rfloor} + (-1)^{\left\lfloor\frac{x}{2}\right\rfloor + \left\lfloor\frac{y}{2}\right\rfloor} + (-1)^{\left\lfloor-\frac{x}{2}\right\rfloor + \left\lfloor-\frac{y}{2}\right\rfloor}\right)$, which is not congruent to $0\pmod{16}$ for $8\nmid n,8\nmid m$. Contradiction, as octomino tiles this rectangle and thus sum covered by all octominoes should be equal to sum covered by rectangle. Thus only assumption we made is false - I octomino doesn't tile $n\times m$ rectangles for $8\nmid n,8\nmid m$. Q.E.D.