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Area: 1.

Perimeter: 4.

Size: 1x1.

Is rectangular: yes.

Is convex: yes.

Holes: 0.

Order: 1.

Square order: 1.

Odd order: 1.

Prime rectangles: 1.

Some facts:

- It is the smallest polyomino.

Smallest rectangle and smallest square and smallest odd rectangle (1x1):

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 |
---|---|---|---|---|---|---|

1 | 11P | |||||

2 | 11C | 11C | ||||

3 | 11C | 11C | 11C | |||

4 | 11C | 11C | 11C | 11C | ||

5 | 11C | 11C | 11C | 11C | 11C | |

6 | 11C | 11C | 11C | 11C | 11C | 11C |

N>0 | all | all | all | all | all | all |

Smallest prime reptile (1Ox2):

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

O monomino | 1 | 1P | 1P | 1C | 1P | 1C |

Smallest torus and smallest square torus and smallest odd torus (1x1):

w \ h | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

1 | 11 | |||||

2 | 11 | 11 | ||||

3 | 11 | 11 | 11 | |||

4 | 11 | 11 | 11 | 11 | ||

5 | 11 | 11 | 11 | 11 | 11 | |

6 | 11 | 11 | 11 | 11 | 11 | 11 |

Smallest Baiocchi figure (area 1):

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

$N(w; h) = 1 \tag{1}$

$T(w; h) = 1 \tag{2}$

$A(w; h) = 1 \tag{3}$

$G(T; x; y) = \frac{xy}{(1-x)(1-y)} \tag{4}$

$G(A; x; y) = \frac{xy}{(1-x)(1-y)} \tag{5}$