Select polyominoes for a set (currently 1 or 2), for which tilings should be shown.

Then click "Show" button.

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

Prime rectangles: ≥ 12.

Smallest rectangle (5x6):

Smallest square (6x6):

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 | 1-4 | 5 | 6 | 7 | 8 | 9 | N>0 |
---|---|---|---|---|---|---|---|

1-4 | 0 | ||||||

5 | 0 | 0 | |||||

6 | 0 | 77P | 3030P | ||||

7 | 0 | 0 | 365365P | 0 | |||

8 | 0 | 227227P | 18321832P | 4149241492P | 373308373308P | ||

9 | 0 | 0 | 1395213952P | 0 | 53119855311985P | 0 | |

10 | 0 | 48204820P | 7515075150C | 30369543036954P | ≥1≥1C | ≥1≥1P | all |

11 | 0 | 0 | 477841477841C | 0 | ≥1≥1C | 0 | 2k |

12 | 0 | 8616286162C | ≥2555≥2555C | ≥133225≥133225C | ≥1≥1C | ≥194658304≥194658304C | all |

13 | 0 | 0 | ≥10950≥10950C | 0 | ≥1≥1C | 0 | 2k |

14 | 0 | 14087691408769C | ≥97664≥97664C | ≥15144580≥15144580C | ≥1≥1C | ≥7.41128147×10¹¹≥74112814720C | all |

N>0 | x | 2k | all | 2k | all | 2k |

Smallest prime reptiles (2Ix5, 5Xx3):

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

I domino | 0 | 0 | 0 | 0 | 4820P |

X pentomino | 0 | 0 | 6144P | 2944512P | ≥1P |

Smallest common multiple (area 20):

area | 10 | 20 |
---|---|---|

solutions | 0 | ≥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 $A(w; h)$

$N(2n + 1; 2m + 1) = T(2n + 1; 2m + 1) = 0 \tag{1}$

Assume domino and X pentomino tile $(2n + 1)\times(2m + 1)$ rectangle. Place numbers in rectangle's cells according to function $F(x,y)\equiv(-1)^{x+y}\pmod{3}$, where $x$ and $y$ are cells' coordinates (zero-based). On the one hand, domino and X pentomino, no matter how placed, cover sum congruent to $0\pmod{3}$. Then sum covered by all polyominoes is also congruent to $0\pmod{3}$. On the other hand, rectangle covers sum congruent to $\sum_{x=0}^{(2n+1)-1}\sum_{y=0}^{(2m+1)-1}(-1)^{x+y}\equiv1\pmod{3}$. Contradiction, as domino and X pentomino tile this rectangle and thus sum covered by all polyominoes should be equal to sum covered by rectangle. Thus only assumption we made is false - domino and X pentomino don't tile $(2n + 1)\times(2m + 1)$ rectangle. Q.E.D.