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

Area: 8.

Size: 3x4.

Holes: 0.

Order: 2.

Square order: 2.

Prime rectangles: ≥ 1.

Smallest rectangle and smallest square (4x4):

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 | 5-7 | 8 | 9-11 | 12 | N>0 |
---|---|---|---|---|---|---|---|

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

4 | 0 | 44P | |||||

5-7 | 0 | 0 | 0 | ||||

8 | 0 | 1616C | 0 | 256256C | |||

9-11 | 0 | 0 | 0 | 0 | 0 | ||

12 | 0 | 6464C | 0 | 40964096C | 0 | 262144262144C | |

13-15 | 0 | 0 | 0 | 0 | 0 | 0 | x |

16 | 0 | 256256C | 0 | 6553665536C | 0 | 1677721616777216C | 4k |

N>0 | x | 4k | x | 4k | x | 4k |

Smallest prime reptile (8Gx4):

polyomino \ n² | 1² | 2² | 3² | 4² | 5² | 6² | 7² | 8² |
---|---|---|---|---|---|---|---|---|

G octomino | 1 | 0 | 0 | 65536P | 0 | 0 | 0 | ≥4294967296P |

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

$N(w; h) = T(w; h) = A(w; h) = 0, \qquad w \not\equiv 0\pmod{4} \tag{1}$

$T(4n; 4m) = 1, \qquad n,m \geq 1 \tag{2}$

$N(4n; 4m) = 4^{nm}, \qquad n,m \geq 1 \tag{3}$

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

$A(4n; 4m) = \sqrt[8]{2}, \qquad n,m \geq 1 \tag{5}$

$G(A; x; y) = \sqrt[8]{2}G(T; x; y) = \frac{\sqrt[8]{2}\,x^4y^4}{(1-x^4)(1-y^4)} \tag{6}$