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Area: 4.

Perimeter: 10.

Size: 2x3.

Is rectangular: no.

Is convex: yes.

Holes: 0.

Order: 2.

Square order: 4.

Odd order: ∞.

Prime rectangles: 2.

Some facts:

- It is smallest rectifiable polyomino which does not tile any rectangle with odd number of pieces.

Smallest rectangle (2x4):

Smallest square (4x4):

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

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

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

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

4 | 0 | 22P | 0 | 1010C | |||||

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

6 | 0 | 0 | 0 | 4242C | 0 | 0 | |||

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

8 | 0 | 44C | 44P | 182182C | 436436C | 43404340C | 1670816708C | 141970141970C | |

9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 611468611468C | 8k |

10 | 0 | 0 | 0 | 790790C | 0 | 0 | 0 | 44170844417084C | 4k |

11 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ≥1≥1C | 8k |

12 | 0 | 88C | 0 | 34323432C | 0 | 517450517450C | 0 | ≥1≥1C | 2k |

13 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ≥1≥1C | 8k |

14 | 0 | 0 | 0 | 1491414914C | 0 | 0 | 0 | ≥1≥1C | 4k |

15 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ≥1≥1C | 8k |

16 | 0 | 1616C | 2424C | 6481464814C | 443144443144C | ≥1≥1C | ≥1≥1C | ≥1≥1C | all |

17 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ≥1≥1C | 8k |

18 | 0 | 0 | 0 | 281680281680C | 0 | 0 | 0 | ≥1≥1C | 4k |

19 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ≥1≥1C | 8k |

20 | 0 | 3232C | 0 | 12241821224182C | 0 | ≥1≥1C | 0 | ≥1≥1C | 2k |

21 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ≥1≥1C | 8k |

22 | 0 | 0 | 0 | 53203105320310C | 0 | 0 | 0 | ≥1≥1C | 4k |

23 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ≥1≥1C | 8k |

24 | 0 | 6464C | 144144C | 2312214823122148C | ≥1≥1C | ≥1≥1C | ≥1≥1C | ≥1≥1C | all |

N>0 | x | 4k | 8k | 2k | 8k | 4k | 8k | all |

Smallest prime reptile (4Lx2):

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

L tetromino | 1 | 4P | 14P | 59009C | 17593273P | ≥267754502500C |

Smallest torus (2x4):

Smallest square torus (4x4):

w \ h | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

1 | 00 | ||||

2 | 00 | 00 | |||

3 | 00 | 00 | 00 | ||

4 | 00 | 3232 | 00 | 992992 | |

5 | 00 | 00 | 00 | 00 | 00 |

6 | 00 | 00 | 00 | 80968096 | 00 |

7 | 00 | 00 | 00 | 00 | 00 |

8 | 00 | 256256 | 864864 | 9792097920 | 275840275840 |

9 | 00 | 00 | 00 | 00 | 00 |

10 | 00 | 00 | 00 | ≥500000≥500000 | 00 |

11 | 00 | 00 | 00 | 00 | 00 |

12 | 00 | 20482048 | 00 | ≥500000≥500000 | 00 |

Smallest Baiocchi figure (area 8):

Smallest Baiocchi figure without holes (area 16):

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

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

Assume L tetromino tiles $n\times m$ rectangles for $nm\not\equiv 0\pmod{8}$.

Place numbers in rectangles' cells according to function $F(x,y)\equiv4x+1\pmod{8}$, where $x$ and $y$ are cells' coordinates (zero-based). On the one hand, L tetromino, no matter how placed, covers sum congruent to $0\pmod{8}$. Then sum covered by all tetrominoes is also congruent to $0\pmod{8}$. On the other hand, rectangle covers sum congruent to $\sum_{x=0}^{n-1}\sum_{y=0}^{m-1}\left(4x+1\right)$, which is not congruent to $0\pmod{8}$ for $nm\not\equiv 0\pmod{8}$. Contradiction, as tetromino tiles this rectangle and thus sum covered by all tetrominoes should be equal to sum covered by rectangle. Thus only assumption we made is false - L tetromino doesn't tile $n\times m$ rectangles for $nm\not\equiv 0\pmod{8}$. Q.E.D.

$T(n; m) = 1, \qquad 8\mid nm \tag{2}$

- Odd rectangle non-existance first proved here: http://www.cflmath.com/~reid/Polyomino/l4_rect.html