Select polyominoes for a set (currently 1 or 2), for which tilings should be shown.
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You may also see list of all polyomino sets for which data is available here.
Area: 5.
Perimeter: 12.
Size: 2x4.
Is rectangular: no.
Is convex: yes.
Holes: 0.
Order: 10.
Square order: 20.
Odd order: 45.
Prime rectangles: ≥ 40.
Smallest rectangle (5x10):
Smallest square (10x10):
Smallest odd rectangle (15x15):
Blue number - strongly prime rectangle (which cannot be divided into two or more number of rectangles tileable by this set).
Green number - weakly prime rectangle (which cannot be divided into two rectangles tileable by this set, but which can be divided into three or more rectangles).
Purple number - prime rectangle (unknown if weakly or strongly prime).
Red number - 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.
Smallest prime reptile (5Yx9):
Smallest torus (2x5):
Smallest square torus (5x5):
Smallest odd torus (5x5):
Smallest Baiocchi figure (area 20):
Smallest known Baiocchi figure without holes (area 40):
$N(w; h)$ - 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 $A(w; h)$
$N(1; n) = T(1; n) = A(1; n) = 0 \tag{1}$
$N(2; n) = T(2; n) = A(2; n) = 0 \tag{2}$
$N(3; n) = T(3; n) = A(3; n) = 0 \tag{3}$
$N(4; n) = T(4; n) = A(4; n) = 0 \tag{4}$
$N(5; n) = 10 \times N(5; n - 10), \quad n \geq 11 \tag{5}$
$N(6; n) = T(6; n) = A(6; n) = 0 \tag{6}$
$N(7; n) = T(7; n) = A(7; n) = 0 \tag{7}$
$N(8; n) = T(8; n) = A(8; n) = 0 \tag{8}$
$G(N(5); x) = \frac{1}{1 - 10x^{10}} \tag{9}$
