Prime packings of the Heptomino P7: Torsten Sillke, FRA Initial version, 1993 --- completed --- x x x x x x x 2-dim: ------ Nx 2, 4, 6, 7, ... {6,7}+2n Zx 2, 4, 6, 7, ... {6,7}+2n 7x 2p, ... 2n 14x 2, 4, 6, 7, ... {6,7}+2n 21x 2, 4, 6, 8, 10, 11p, ... {10,11}+2n 28x 2, 4, 6, 7, ... {6,7}+2n 35x 2, 4, 6, 8, 10, 11, ... {10,11}+2n There is one further prime-rectangle of width 9. In Anton Hanegraaf's list [2] there is a 9x63 mentinoned. This has been found manually by Andrew Clarke [1]. Later Helmut Postl [2] found a 9x77 manually. In the meantime he found a 9x63 too and ruled out the 9x49. 3-dim: ------ 2x7x 1p, ... n 3x7x 2, 4, 5p, ... {4,5}+2n Impossible: Zx{3,5} 7xk with k odd 9x{21, 35} 3x3x..x3xN (you can't fill the 3x3x..x3 border) Annotaions: the 7xN strip dissects into 2x7 and 4x7 blocks. a a a a a a a a a a a b a a c c b b a b c c b b b b c c b b b b c d d d b b d d d d the 3x5x7 box: b c c d d with a,b,c,d are 2x7 blocks b 7 7 7 7 a a 7 7 7 dist2 coloring: x . x . x . x . x . x . x . x . x . x . x . . . . . . . . . . . . . . . . . . . . . x . x . x . x . x . x . x . x . x . x . x . . . . . . . . . . . . . . . . . . . . . x . x . x . x . x . x . x . x . x . x . x . . . . . . . . . . . . . . . . . . . . . x . x . x . x . x . x . x . x . x . x . x . . . . . . . . . . . . . . . . . . . . . x . x . x . x . x . x . x . x . x . x . x . . . . . . . . . . . . . . . . . . . . . x . x . x . x . x . x . x . x . x . x . x This coloring shows that 11x21 may be the smalles odd rectangle. The number of 'x' is 6*11 = 66. Each P7 fits at most 2. As there are 33 P7 each must fit exactly 2 'x'. the 11x21 rectangle: X X 39 39 39 39 Y Y Y Y Y Y Y 22 22 22 22 15 15 15 15 X X 40 39 39 39 Y Y Y Y Y Y Y 22 22 22 19 16 15 15 15 X X 40 40 Z Z Z Z Z Z Z 26 26 23 23 19 19 16 16 13 13 X X 40 40 Z Z Z Z Z Z Z 26 26 23 23 19 19 16 16 13 13 X X 40 40 38 38 38 32 32 32 32 26 26 23 23 19 19 16 16 13 13 X X 41 38 38 38 38 34 32 32 32 27 26 23 21 21 18 18 18 18 13 X X 41 41 37 37 34 34 30 30 27 27 25 25 21 21 18 18 18 14 14 43 43 41 41 37 37 34 34 30 30 27 27 25 25 21 21 20 20 20 14 14 43 43 41 41 37 37 34 34 30 30 27 27 25 25 21 20 20 20 20 14 14 43 43 42 42 42 37 35 35 35 30 29 29 29 25 24 24 24 17 17 17 14 43 42 42 42 42 35 35 35 35 29 29 29 29 24 24 24 24 17 17 17 17 the 9x66 rectangle: solution of [1] 56 56 56 56 50 50 50 50 45 45 45 45 . 3 solutions for the end-parts 56 56 56 54 51 50 50 50 46 45 45 45 . 57 57 54 54 51 51 49 49 46 46 44 44 . 57 57 54 54 51 51 49 49 46 46 44 44 . 57 57 54 54 51 51 49 49 46 46 44 44 . 58 57 55 53 53 53 53 49 47 47 47 44 43 58 58 55 55 53 53 53 47 47 47 47 43 43 58 58 55 55 X X X X X X X 43 43 58 58 55 55 X X X X X X X 43 43 56 56 56 56 51 51 51 51 46 46 46 46 . 56 56 56 53 51 51 51 48 47 46 46 46 . 57 57 53 53 52 52 48 48 47 47 44 44 . 57 57 53 53 52 52 48 48 47 47 44 44 . 57 57 53 53 52 52 48 48 47 47 44 44 . 58 57 54 54 54 52 49 49 49 49 45 44 43 58 58 54 54 54 54 49 49 49 45 45 43 43 58 58 X X X X X X X 45 45 43 43 58 58 X X X X X X X 45 45 43 43 56 56 56 56 51 51 51 51 46 46 46 46 . 57 56 56 56 51 51 51 48 47 46 46 46 . 57 57 53 53 53 53 48 48 47 47 44 44 . 57 57 54 53 53 53 48 48 47 47 44 44 . 57 57 54 54 52 52 48 48 47 47 44 44 . 58 58 54 54 52 52 49 49 49 49 45 44 43 58 58 54 54 52 52 49 49 49 45 45 43 43 58 58 55 55 55 52 50 50 50 45 45 43 43 58 55 55 55 55 50 50 50 50 45 45 43 43 57 57 57 57 51 51 51 51 46 46 46 46 41 41 41 41 36 36 36 36 30 30 30 30 25 25 25 25 21 21 19 19 16 16 14 14 11 11 . 57 57 57 55 52 51 51 51 47 46 46 46 42 41 41 41 37 36 36 36 31 30 30 30 26 25 25 25 21 21 19 19 16 16 14 14 11 11 . 58 58 55 55 52 52 49 49 47 47 44 44 42 42 39 39 37 37 34 34 31 31 29 29 26 26 24 24 21 21 19 19 16 16 14 14 11 11 . 58 58 55 55 52 52 49 49 47 47 44 44 42 42 39 39 37 37 34 34 31 31 29 29 26 26 24 24 21 20 19 18 16 15 14 12 11 10 . 58 58 55 55 52 52 49 49 47 47 44 44 42 42 39 39 37 37 34 34 31 31 29 29 26 26 24 24 23 20 20 18 18 15 15 12 12 10 10 . 58 56 53 53 53 49 48 48 48 44 43 43 43 39 38 38 38 34 33 33 33 29 28 28 28 24 23 23 20 20 18 18 15 15 12 12 10 10 . 56 56 53 53 53 53 48 48 48 48 43 43 43 43 38 38 38 38 33 33 33 33 28 28 28 28 23 23 20 20 18 18 15 15 12 12 10 10 . 56 56 D D D D D D D Y Y Y Y Y Y Y D D D D D D D 27 27 27 23 23 22 22 22 X X X X X X X . 56 56 D D D D D D D Y Y Y Y Y Y Y D D D D D D D 27 27 27 27 22 22 22 22 X X X X X X X The the 2x7 blocks which contain a defective P7 according the dist2 coloration are marked 'D'. References: [1] Andrew Clarke; letter to Jean Meeus, 1974-03-03 [2] Anton Hanegraaf; Summary of known prime boxes for solid polyominoes, (version from 1995-06-08) [3] Helmut Postl; letter to Torsten Sillke, 1997-07-23