Prime Packings of the Pentomino F. Torsten Sillke, FRA, 1993 >>>>>>>>>>> 3D complete <<<<<<<<<< update: no 5x5x9 22.03.96 update: 5x5xk k>=13 22.03.96 update 5x5x11 26.03.96 update 5x8x8, 5x8x9 15.04.96 update 3x11x15 16.04.96 update 4x{7,8,9}x15 16.04.96 update 5x7x9 24.04.96 update 5x7x11 29.04.96 update 5x9x11 04.05.96 update 5x9x9 06.05.96 Johan van de Konijnenberg update no 5x7x7 01.09.08 update 5x7x13 10.10.08 2-dim: ------ Zx 5p, ... only 5n 3-dim: ------ 4x5x 10p, 12p, 13p, 14p, 15p, 16p, 17p, 18p, 19p, 20, 21p, ... {12..21} + 10n 5x5x 8p, 10p, 11p, 12p, 13p, 14p, 15p, 16, 17p, ... {10..17} + 8n 6x5x 6p, 7p, 8p, 9p, 10, 11p, ... {6..11} + 6n 7x5x 6p, 8p, 9p, 10p, 11p, 12, 13p, ... {8..13} + 6n 8x5x 5p, 6p, 7p, 8p, 9p, ... {5..9} + 5n 9x5x 6p, 7p, 8p, 9p, 10, 11p, ... {6..11} + 6n 10x5x 4p, 5p, 6, 7p, ... {4..7} + 4n 11x5x 5p, 6p, 7p, 8, 9p, ... {5..9} + 5n 3x10x 6p, 7p, 8p, 9p, 10p, 11p, ... {6..11} + 6n 4x10x 5p, 7p, 8p, 9p, 10, 11p, ... {7..11} + 5n 5x10x 4p, 5p, 6, 7p, ... {4..7} + 4n 3x15x 6p, 7p, 8p, 9p, 10, 11p, ... {6..11} + 6n 4x15x 5p, 6p, 7p, 8p, 9p, ... {5..9} + 5n 5x15x 4p, 5p, 6, 7, ... {4..7} + 4n Impossible: NxN, 3x{3,4,5}xN, 4x{3,4}xN, 4x5x{3..9,11}, 5x5x{3..7,9}, 5x7x7 Annotations: 4x5x10: 47 48 38 39 29 29 23 23 18 15 48 48 48 39 39 29 29 18 18 18 48 49 39 39 33 29 25 18 12 19 49 49 49 33 33 25 25 19 19 19 49 46 46 37 33 33 25 25 19 17 47 44 38 34 32 23 23 15 15 15 47 40 38 32 32 32 24 16 12 10 47 41 38 32 28 24 24 24 12 10 45 41 41 41 28 28 28 24 12 11 46 46 41 37 37 28 22 17 17 17 44 44 44 34 34 27 23 16 15 10 47 40 38 36 27 27 27 16 12 10 45 40 36 36 36 31 27 16 20 11 45 40 36 31 31 31 20 20 20 11 45 46 37 37 31 22 22 20 17 11 44 42 34 34 30 30 21 21 13 13 42 42 42 30 30 21 21 13 13 10 43 40 42 35 30 26 21 16 13 14 43 43 43 35 35 26 26 14 14 14 45 43 35 35 26 26 22 22 14 11 3x10x10 (symmetric): 34 24 24 17 2 2 2 2 2 2 35 25 24 24 11 2 2 2 2 2 36 26 24 21 14 2 2 2 2 2 36 36 36 18 14 14 2 2 2 2 37 36 27 14 14 2 2 2 2 2 37 27 27 27 10 2 2 2 2 2 38 27 22 15 10 2 2 2 2 2 38 28 29 29 16 2 2 2 2 2 39 29 29 16 16 2 2 2 2 2 39 33 29 20 16 16 2 2 2 2 34 34 34 17 17 2 2 2 2 2 35 25 25 25 11 11 2 2 2 2 35 26 21 21 2 2 2 2 2 2 37 26 23 18 18 2 2 2 2 2 37 26 23 23 10 2 2 2 2 2 38 23 23 15 10 2 2 2 2 2 38 28 22 15 13 2 2 2 2 2 39 28 22 13 13 13 2 2 2 2 39 28 22 13 2 2 2 2 2 2 33 33 33 20 20 2 2 2 2 2 35 34 17 17 2 2 2 2 2 2 35 30 25 11 11 2 2 2 2 2 30 30 30 21 21 2 2 2 2 2 30 26 18 18 19 2 2 2 2 2 37 31 19 19 19 2 2 2 2 2 31 31 31 19 10 2 2 2 2 2 38 28 31 15 2 2 2 2 2 2 32 32 22 15 12 2 2 2 2 2 39 32 32 12 12 2 2 2 2 2 33 32 20 20 12 12 2 2 2 2 5x5x8: 67 67 53 53 2 2 2 2 68 67 67 53 53 2 2 2 68 67 60 53 2 2 2 2 69 60 60 2 2 2 2 2 69 66 60 60 2 2 2 2 68 64 57 2 2 2 2 2 68 57 57 57 52 2 2 2 69 58 59 57 2 2 2 2 69 59 59 59 2 2 2 2 66 66 66 59 2 2 2 2 64 64 64 54 2 2 2 2 68 58 52 52 52 2 2 2 65 58 56 56 2 2 2 2 69 56 56 2 2 2 2 2 66 63 56 51 2 2 2 2 64 54 54 54 2 2 2 2 65 55 55 52 2 2 2 2 65 58 55 55 2 2 2 2 65 58 55 2 2 2 2 2 63 63 63 51 51 2 2 2 61 61 54 50 2 2 2 2 65 61 61 50 50 2 2 2 62 61 50 50 2 2 2 2 62 62 62 2 2 2 2 2 63 62 51 51 2 2 2 2 5x5x10: 5 4 5 5 6 6 5 5 4 5 = X (symmetric) 5 5 5 5 5 5 6 5 5 4 2 X -> 5x5x10 4 5 5 6 5 37 30 30 20 20 38 35 30 30 20 20 38 39 30 23 20 39 39 39 23 23 39 36 23 23 . 37 37 37 18 . 35 35 35 18 15 38 28 29 19 19 38 29 29 29 19 19 36 36 36 29 19 32 37 22 22 15 33 28 35 18 15 38 28 27 18 15 34 28 27 27 21 36 27 27 26 17 32 32 32 22 22 33 28 25 18 . 33 25 25 25 15 34 25 21 21 21 34 26 26 26 17 17 33 32 24 22 16 16 33 24 24 16 16 34 31 24 24 16 34 31 31 21 . 31 31 26 17 17 5x5x12: 88 88 74 74 66 66 2 2 2 2 89 88 88 74 74 67 2 2 2 2 89 88 80 74 67 67 67 2 2 2 ... 90 80 80 79 67 63 2 2 2 2 90 87 80 80 71 71 2 2 2 2 89 85 77 77 65 66 66 2 2 2 89 77 77 76 70 70 2 2 2 2 90 78 77 70 70 63 2 2 2 2 90 79 79 79 70 63 2 2 2 2 87 87 87 71 71 63 2 2 2 2 85 85 85 72 65 66 2 2 2 2 89 76 76 76 65 61 2 2 2 2 86 78 78 78 65 61 2 2 2 2 90 82 79 69 64 62 2 2 2 2 87 84 73 73 71 63 2 2 2 2 85 83 83 72 72 61 2 2 2 2 83 83 76 69 65 61 2 2 2 2 86 83 78 69 64 62 2 2 2 2 86 82 82 69 64 62 2 2 2 2 84 84 84 73 73 2 2 2 2 2 81 81 72 72 68 68 2 2 2 2 86 81 81 68 68 61 2 2 2 2 86 81 75 69 68 2 2 2 2 2 82 82 75 75 64 62 2 2 2 2 84 75 75 73 64 62 2 2 2 2 5 4 4 4 4 4 4 4 A has 4 Solutions 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 = A ok 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 A and A -> 5x8x8 6 5 5 5 5 5 5 5 A and B -> 5x8x9 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 = B ok 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 4 6 6 6 5 5 5 5 6 6 6 5 5 5 5 6 6 6 5 5 5 5 = C ok 6 6 6 5 5 5 5 6 6 6 5 5 5 5 6 6 6 6 5 5 5 C and D -> 5x7x11 6 6 6 6 5 5 5 6 6 6 6 5 5 5 = D ok 6 6 6 6 5 5 5 6 6 6 6 5 5 5 6 6 6 6 5 5 5 5 5 E and F -> 5x9x11 6 6 6 6 5 5 5 5 5 6 6 6 6 5 5 5 5 5 = E ok 6 6 6 6 5 5 5 5 5 6 6 6 6 5 5 5 5 5 6 6 6 6 6 5 5 5 5 6 6 6 6 6 5 5 5 5 6 6 6 6 6 5 5 5 5 = F ok 6 6 6 6 6 5 5 5 5 6 6 6 6 6 5 5 5 5 4 4 3 3 4 4 4 4 4 G and G -> 5x8x9 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 = G ok 4 4 4 4 4 4 4 4 4 4 4 5 5 4 4 4 4 4 5 5 4 4 5 5 5 5 5 G and H -> 5x9x9 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 = H ok 5 5 5 5 5 5 5 5 5 5 5 6 6 5 5 5 5 5 4 4 4 4 4 3 3 4 4 I and I -> 5x8x9 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 = I ok 4 4 4 4 4 4 4 4 4 4 4 5 5 4 4 4 4 4 4 4 4 4 3 3 4 4 4 J and J -> 5x8x9 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 = J ok 4 4 4 4 4 4 4 4 4 4 4 4 5 5 4 4 4 4 extensions n -> n+1 x x - - - - x x X X - - with reflected back part 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 - - - 5x5xk Boxes for k>=13 for the Pentomino F. Torsten Sillke 22.03.96 62 54 54 . . . . . . . . 5x5xk front part 63 55 54 54 . . . . . . . 63 64 54 49 . . . . . . . 64 64 64 49 49 . . . . . . 64 61 49 49 . . . . . . . . . . . . . . . . . . 62 62 62 48 . . . . . . . 60 55 52 48 48 . . . . . . 63 55 48 48 44 . . . . . . 63 55 53 44 44 44 . . . . . 61 61 61 44 47 . . . . . . . . . . . . . . . . . 60 62 58 46 42 . . . . . . 60 52 52 46 46 40 . . . . . 63 55 46 46 40 40 . . . . . 59 53 53 53 43 40 40 . . . . 61 57 47 47 47 39 . . . . . . . . . . . . . . . . 58 58 58 42 42 38 . . . . . 60 51 52 52 38 38 38 . . . . 60 51 51 51 38 37 37 . . . . 59 53 51 43 43 43 37 37 . . . 59 57 57 47 39 39 37 . . . . . . . . . . . . . . . 56 58 45 45 42 42 36 . . . . 56 56 56 45 45 36 36 36 . . . 59 56 50 45 41 41 35 36 . . . 59 50 50 41 41 43 35 35 35 . . 57 57 50 50 41 39 39 35 . . . 62 63 53 53 43 . . . . . . 5x5xk back part 63 63 63 45 . . . . . . . 63 64 49 50 40 . . . . . . 64 64 64 50 50 . . . . . . 64 61 50 50 44 44 . . . . . . . . . . . . . . . . 62 53 53 43 43 43 . . . . . 62 54 49 45 40 . . . . . . 62 58 49 45 40 39 . . . . . 60 60 49 45 39 39 . . . . . 61 61 61 44 44 39 39 . . . . . . . . . . . . . . . 57 54 53 52 41 43 33 . . . . 62 54 49 46 46 36 . . . . . 58 58 58 45 40 37 35 . . . . 59 60 60 47 40 35 35 . . . . 61 56 48 48 44 38 35 35 . . . . . . . . . . . . . . 57 52 52 52 41 41 33 33 . . . 57 54 46 46 42 36 36 . . . . 57 54 58 42 42 37 37 32 . . . 59 60 47 47 42 42 32 32 . . . 59 56 56 48 48 38 38 32 32 . . . . . . . . . . . . . 55 55 52 41 41 33 33 31 31 . . 57 55 55 46 36 36 31 31 . . . 59 55 51 51 37 37 34 31 30 . . 59 51 51 47 47 34 34 30 30 . . 56 56 51 48 38 38 34 34 30 30 . 67 67 56 49 43 43 40 29 29 2 2 Box 5x5x11 with the F-Pentomino 68 67 67 49 49 49 40 40 29 29 2 68 67 60 50 49 40 40 33 29 2 2 Torsten Sillke, 26.03.96 69 60 60 50 50 50 36 36 2 2 2 69 66 60 60 50 42 42 35 35 2 2 (inner part symmetric) 68 58 56 56 41 43 43 31 2 2 2 68 58 59 53 41 41 41 32 2 2 2 69 59 59 59 54 41 33 33 33 2 2 69 59 54 54 54 48 34 36 36 25 2 66 66 66 54 42 42 35 35 2 2 2 64 56 56 46 47 43 31 31 31 2 2 68 58 53 53 47 47 32 32 32 28 2 65 58 57 47 47 39 33 28 28 28 2 69 57 57 48 48 48 34 36 28 25 2 66 62 57 57 45 42 34 35 26 25 2 64 64 64 46 46 46 31 27 2 2 2 65 58 52 53 53 39 32 27 27 27 2 65 63 52 52 52 39 34 30 27 25 2 63 63 63 52 48 39 34 30 30 25 2 63 62 62 45 45 45 30 30 26 26 2 61 64 51 51 46 37 37 2 2 2 2 61 61 61 51 51 39 37 37 2 2 2 65 61 55 51 44 44 37 38 2 2 2 65 55 55 44 44 38 38 38 2 2 2 62 62 55 55 44 45 38 26 26 2 2 5x7x13: 66 66 48 48 37 37 . 17 18 18 2 67 66 66 48 48 . 30 18 18 2 2 67 66 49 48 . 34 34 26 18 2 2 68 56 49 49 49 . 34 34 2 2 2 68 57 50 49 40 . 34 31 2 2 2 69 57 57 57 40 40 . 31 31 14 2 69 65 57 40 40 . 31 31 29 14 2 67 62 55 37 37 . 30 17 17 17 2 67 61 55 55 38 . 30 25 25 2 2 68 55 55 38 38 . 30 26 26 2 2 68 56 50 53 38 38 . 27 28 2 2 69 56 50 39 47 . 28 28 28 14 2 69 56 47 47 47 36 . 28 15 14 2 65 65 65 47 46 . 29 29 29 2 2 62 62 62 41 37 . 30 21 17 2 2 67 61 61 45 . 33 25 25 2 2 2 63 52 45 45 45 . 26 26 2 2 2 68 53 53 53 45 . 27 27 27 2 2 64 56 50 39 39 39 . 23 15 2 2 69 54 50 36 36 36 . 24 15 14 2 65 60 46 46 46 . 32 29 15 2 2 62 51 51 41 41 . 21 21 21 2 2 61 61 51 51 . 33 22 25 2 2 2 63 52 51 44 . 33 22 22 22 2 2 63 52 53 44 44 . 27 22 2 2 2 64 52 44 44 39 . 23 23 23 2 2 64 54 54 54 36 . 24 24 24 2 2 64 60 60 46 . 32 32 32 15 2 2 58 58 41 41 . 33 21 19 19 2 2 63 58 58 42 . 33 19 19 2 2 2 63 58 42 42 42 35 . 19 16 2 2 59 52 42 35 35 35 . 16 16 2 2 59 59 59 43 35 . 23 20 16 16 2 64 59 54 43 43 . 24 20 20 2 2 60 60 43 43 . 32 20 20 2 2 2 tiling the 5xZ strip: (two possibilities) 1 1 2 2 3 3 4 4 1 1 2 2 3 3 4 4 1 5 2 6 3 7 4 8 5 5 6 6 7 7 8 8 5 5 6 6 7 7 8 8 1 1 2 2 3 3 4 4 1 1 2 2 3 3 4 4 1 5 2 6 3 7 4 8 5 5 6 6 7 7 8 8 5 5 6 6 7 7 8 8 tiling the plane: There are other possibilities than 5xZ strips. 3 3 3 3 3 2 2 3 3 3 2 2 2 3 3 1 1 2 2 2 1 1 1 2 2 1 1 1 1 1