prime boxes of (81) Torsten Sillke, 05.02.93 Update: 5x5x6, 13.02.96 Update: 2x10x10, 2x9x15, 17.06.98 Update: 3x4x30 18.06.98 Update: 4x4x10 22.06.98 Update: 3x15x{7,8} 09.07.98 H Postl Completed 3 dim at 1998-07-09 by H Postl 1 2 1 1 3x5x 6p, 9p, ... {6, 9}+6n 4x5x 6p, ... 6n 5x5x 6p, ... 6n 6x5x 3p, 4p, 5p, ... {3..5}+3n 2x10x 8p, 10p, 12p, 14p, ... {8, 10, 12, 14}+8n 3x10x 6, 9, ... {6, 9}+6n 4x10x 4p, 6, ... {4,6}+4n 2x15x 9p, 12p, 15p, ... {9,12,15}+9n 3x15x 5, 6, 7p, 8p, 9, ... {5..9}+5n 2x20x 8, 10, 12, 14, ... {8, 10, 12, 14}+8n 3x20x 6, 9, ... {6, 9}+6n 4x20x 4, 6, ... {4, 6}+4n 2x25x 12, 18, ... {12, 18}+12n 3x25x 6, 9, ... {6, 9}+6n 2x30x 8, 9, 10, 11p, 12, 13p, 14, 15, ... {8..15}+8n 3x30x 4p, 5, 6, 7, ... {4..7}+4n 4x30x 3p, 4, 5, ... {3..5}+3n 2xNx 8, 9, 10, 11, 12, 13, 14, 15, ... {8..15}+8n 3xNx 4, 5, 6, 7 ... {4..7}+4n 4xNx 3, 4, 5, ... {3..5}+3n 5xNx 3, 4, 5, ... {3..5}+3n 6xNx 3, 4, 5, ... {3..5}+3n Complete list of prime boxes (# = 18): 2x8x10, 2x9x15, 2x10x{10,12,14}, 2x11x30, 2x12x15, 2x13x30, 2x15x15, 3x4x{30,45}, 3x5x{6,9}, 3x7x15, 3x8x15, 4x4x10, 4x5x6, 5x5x6. Impossible: 2xNx{2,3,4,5,6,7} 3xNx3 3x4x15 uxpxq with u odd and 0 != p*q (mod 3) Thm[1]: A uxpxq box with u odd has the property 3|p*q. Proof: color the cube (x,y,z) with x (modulo 2). The piece covers 1 (mod 3) number of cubes of each color. Therefore both color must be equal (mod 3) and that is 3|p*q. Annotations: Ring symmetry: 2 1 1 1 . 1 1 1 2 the 2*10*10 box: 99 93 89 84 84 84 71 71 64 64 98 93 93 93 78 84 79 71 64 68 98 92 88 83 78 75 75 71 64 67 98 98 88 87 78 78 75 74 63 63 97 97 88 88 77 82 75 66 63 62 97 90 90 82 82 82 70 66 63 60 97 91 90 81 76 76 76 66 66 60 96 94 90 81 86 73 76 72 60 60 95 94 85 81 81 72 72 72 61 65 94 94 85 85 85 80 69 65 65 65 99 99 89 89 89 84 79 71 64 68 99 93 89 83 79 79 79 68 68 68 99 92 92 83 83 83 75 67 67 67 98 92 87 87 78 74 74 74 63 67 97 92 88 87 77 77 70 74 62 62 96 91 90 87 77 82 70 70 70 62 96 91 91 91 77 73 76 66 61 62 96 96 86 86 86 73 73 73 61 60 95 95 95 81 86 80 69 72 61 61 95 94 85 80 80 80 69 69 69 65 the 2*9*15 box: 98 98 90 90 81 81 78 2 2 2 2 2 2 2 2 99 98 90 86 81 80 75 2 2 2 2 2 2 2 2 97 98 90 85 81 80 75 2 2 2 2 2 2 2 2 96 96 96 88 80 80 75 75 2 2 2 2 2 2 2 95 95 96 88 89 77 77 73 2 2 2 2 2 2 2 95 94 88 88 83 77 79 73 2 2 2 2 2 2 2 95 91 93 84 83 77 73 73 2 2 2 2 2 2 2 92 91 87 83 83 76 76 76 2 2 2 2 2 2 2 91 91 87 87 87 82 74 76 2 2 2 2 2 2 2 99 98 90 86 81 78 78 2 2 2 2 2 2 2 2 99 99 99 86 86 86 78 2 2 2 2 2 2 2 2 97 97 97 85 85 85 78 2 2 2 2 2 2 2 2 97 94 96 85 89 80 75 2 2 2 2 2 2 2 2 95 94 89 89 89 77 79 2 2 2 2 2 2 2 2 94 94 93 88 79 79 79 2 2 2 2 2 2 2 2 93 93 93 84 84 84 74 73 2 2 2 2 2 2 2 92 92 92 84 83 82 74 76 2 2 2 2 2 2 2 92 91 87 82 82 82 74 74 2 2 2 2 2 2 2 the 4*4*10 box: 98 94 94 87 85 85 2 2 2 2 99 95 94 87 87 85 2 2 2 2 99 95 95 88 86 85 2 2 2 2 99 99 93 88 88 84 2 2 2 2 98 98 94 89 86 85 2 2 2 2 98 92 93 87 86 84 2 2 2 2 98 95 93 86 86 84 2 2 2 2 99 93 93 88 84 84 2 2 2 2 96 96 94 89 2 2 2 2 2 2 96 92 91 87 2 2 2 2 2 2 96 95 91 90 2 2 2 2 2 2 97 91 91 88 2 2 2 2 2 2 96 92 89 89 2 2 2 2 2 2 97 92 92 89 2 2 2 2 2 2 97 90 90 90 2 2 2 2 2 2 97 97 91 90 2 2 2 2 2 2 the 3*4*30 box: (point symmetric) 97 93 93 93 90 83 83 83 78 78 78 71 71 66 66 2 2 2 2 2 98 98 98 93 90 89 84 83 79 81 78 71 74 66 2 2 2 2 2 2 ... 99 94 98 90 90 84 84 81 81 81 74 74 74 66 2 2 2 2 2 2 97 97 97 93 87 87 87 83 77 77 78 71 68 66 2 2 2 2 2 2 97 94 98 89 89 89 87 80 79 77 73 70 70 70 2 2 2 2 2 2 99 94 94 94 90 89 84 80 80 81 73 73 74 70 2 65 2 2 2 2 O 92 91 91 91 86 87 82 79 77 72 71 68 68 2 64 2 2 2 2 99 92 92 92 91 86 85 79 79 75 75 75 68 70 67 65 2 2 2 2 99 99 O 88 86 86 84 80 76 75 73 69 68 65 65 65 2 2 2 2 O O O 88 91 82 82 82 76 77 72 72 72 64 64 64 2 2 2 2 O 92 O 88 85 85 85 82 76 75 72 69 67 67 67 64 2 2 2 2 O O O 88 88 86 85 80 76 76 73 69 69 69 67 2 2 2 2 2 the 4*5*6 box: 67 61 57 52 47 47 67 67 57 52 52 47 68 62 57 57 48 48 68 62 62 62 58 48 68 68 58 58 58 48 65 61 61 61 50 47 67 61 59 52 50 45 66 66 57 50 50 48 66 62 56 51 51 51 68 63 56 56 58 51 65 60 55 55 55 47 67 60 59 52 55 45 66 60 60 53 50 46 64 64 64 49 54 51 64 63 56 49 49 49 65 65 59 53 55 45 65 59 59 53 45 45 66 60 53 53 46 46 64 63 54 54 54 46 63 63 56 49 54 46 the 5*5*6 box: 88 81 75 69 69 69 88 88 75 75 75 69 89 82 82 82 74 63 89 82 76 76 76 63 89 89 76 80 63 63 86 81 81 81 72 69 88 81 75 68 68 68 87 82 74 74 74 68 87 87 76 80 74 67 89 80 80 80 62 63 86 78 72 72 72 61 88 78 78 66 72 68 84 79 79 66 66 66 87 79 73 67 67 67 85 79 73 73 62 67 86 86 83 64 61 61 86 78 70 70 70 61 84 79 71 66 70 61 87 77 77 77 62 60 85 77 73 65 62 62 83 83 83 64 64 64 84 78 83 64 70 60 84 84 71 71 71 60 85 77 71 65 60 60 85 85 73 65 65 65 References: [1] H. Postl many new prime packings. letter from 9. July 1998