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Area: 20.

Perimeter: 18.

Size: 3x10.

Is rectangular: no.

Is convex: yes.

Holes: 0.

Order: 12.

Square order: 20.

Prime rectangles: ≥ 3.

Smallest rectangle (12x20):

Smallest square (20x20):

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 | 1-11 | 12 | 13-15 | 16 | 17-19 | 20 | N>0 |
---|---|---|---|---|---|---|---|

1-11 | 0 | ||||||

12 | 0 | 0 | |||||

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

16 | 0 | 0 | 0 | 0 | |||

17-19 | 0 | 0 | 0 | 0 | 0 | ||

20 | 0 | 22P | 0 | 22P | 0 | 22P | |

21 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

22 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

23 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

24 | 0 | 0 | 0 | 0 | 0 | 66C | ? |

25 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

26 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

27 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

28 | 0 | 0 | 0 | 0 | 0 | 1010C | ? |

29 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

30 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

31 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

32 | 0 | 0 | 0 | 0 | 0 | 1414C | ? |

33 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

34 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

35 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

36 | 0 | 0 | 0 | 0 | 0 | 2626C | ? |

37 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

38 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

39 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

40 | 0 | 44C | 0 | 44C | 0 | 4646C | ? |

41 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

42 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

43 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

44 | 0 | 0 | 0 | 0 | 0 | 7474C | ? |

45 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

46 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

47 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

48 | 0 | 0 | 0 | 0 | 0 | 126126C | ? |

49 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

50 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

51 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

52 | 0 | 0 | 0 | 0 | 0 | 218218C | ? |

53 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

54 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

55 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

56 | 0 | 0 | 0 | 0 | 0 | 366366C | ? |

57 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

58 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

59 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

60 | 0 | 88C | 0 | 88C | 0 | 618618C | ? |

61 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

62 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

63 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

64 | 0 | 0 | 0 | 0 | 0 | 10541054C | ? |

65 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

66 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

67 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

68 | 0 | 0 | 0 | 0 | 0 | 17861786C | ? |

69 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

70 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

71 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

72 | 0 | 0 | 0 | 0 | 0 | 30223022C | ? |

73 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

74 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

75 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

76 | 0 | 0 | 0 | 0 | 0 | 51305130C | ? |

77 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

78 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

79 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

80 | 0 | 1616C | 0 | 1616C | 0 | 87028702C | ? |

81 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

82 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

83 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

84 | 0 | 0 | 0 | 0 | 0 | 1474614746C | ? |

85 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

86 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

87 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

88 | 0 | 0 | 0 | 0 | 0 | 2500625006C | ? |

89 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

90 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

91 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

92 | 0 | 0 | 0 | 0 | 0 | 4241042410C | ? |

93 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

94 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

95 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

96 | 0 | 0 | 0 | 0 | 0 | 7190271902C | ? |

97 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

98 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

99 | 0 | 0 | 0 | 0 | 0 | 00 | ? |

100 | 0 | 3232C | 0 | 3232C | 0 | 121914121914C | ? |

N>0 | x | 20k | x | 20k | x | ? |

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

$N(12; n) = 2 \times N(12; n - 20), \qquad n \geq 21 \tag{1}$

$N(16; n) = 2 \times N(16; n - 20), \qquad n \geq 21 \tag{2}$