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.
Area: 6.
Size: 2x4.
Holes: 0.
Order: 2.
Square order: 24.
Prime rectangles: ≥ 1.
Smallest rectangle (3x4):
Smallest square (12x12):
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 (6Fx8):
N(w;h) - number of ways to tile w×h rectangle (including symmetric solutions)
T(w;h)={1,N(w;h)≥10,else - tileability function, 1 if tiles rectangle, 0 otherwise
A(w;h)=(N(w;h))1wh - average number of ways to tile cell in w×h rectangle (including symmetric solutions)
G(T;x;y)=∑∞w=1∑∞h=1T(w;h)xwyh - bivariate generating function of T(w;h)
G(A;x;y)=∑∞w=1∑∞h=1A(w;h)xwyh - bivariate generating function of A(w;h)
N(1;n)=T(1;n)=0,n≥1
N(2;n)=T(2;n)=0,n≥1
N(3;n)=2×N(3;n−4),n≥5
N(4;n)=2×N(4;n−3),n≥4
N(5;n)=T(5;n)=0,n≥1
N(6;n)=4×N(6;n−4),n≥5
N(7;n)=288×N(7;n−12),n≥13
N(8;n)=4×N(8;n−3),n≥4
N(9;n)=8×N(9;n−4),n≥5
N(10;n)=4384×N(10;n−12)−483584×N(10;n−24)+3604480×N(10;n−36)+100663296×N(10;n−48),n≥49
N(11;n)=7296×N(11;n−12),n≥13
N(12;n)=8×N(12;n−3)+20×N(12;n−4)−64×N(12;n−8),n≥9