prime boxes of (21) Torsten Sillke, FRA initial Version 1993 Update: 4x10x10; 4x8x20; 6x6x15,20,25; 6x7x10,15 H Postl[1] Update: no 4x8x10; no 5x6x{6,7}; no 6x6x10 H Postl[1] Update: no 5x6x{9,10,11,13,14,15}; no 5x7x{12,18} 1998-08-12 Update: 5x7x{24,30,36} 1998-08-13 Update: 5x7x42; 4x10x14; 4x9x15 1998-08-14 1 1 1 2 4x5x 6p, ... 6n 6x5x 4p, 8, 12, 16, ... (only 4n boxes ???) 7x5x 24p, 30p, 36p, 42p ... {24,30,36,42}+24n 4x10x 6, 10p, 12, 14p, ... {10,12,14}+6n 5x10x 12?, 18? ... {12,18}+12n 6x10x 4, 7p, 8, 9?, 10?, ... {7..10}+4n 7x10x 6p, ... 6n 4x15x 6, 9p ... {6,9}+6n 6x15x 4, 6p, 7p, 8, 9?, ... {6..9}+4n 4x20x 6, 8p, 10, ... {6,8,10}+6n 5x20x 6, ... 6n 6x20x 4, 5, 6p, 7 ... {4..7}+4n 6x25x 4, 6p, 7, 8, 9?, ... {6..9}+4n 4x30x 5, 6, 8? Impossible: 2xNx{2..20} 3xNx{2..12} 4xNx{4,7} 5xNx{5} 4x8x10, 5x6x{6,7,9,10,11,13,14,15} 5x7x{12,18} 6x6x10 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: the 5x6xN strip: conjecture: there is no 5x6x2u box with u odd. corollary: all boxes are of the form 5x6x4k. Indicators: After placing 12 pieces only 23 configurations are 100-extendable. After placing 24 pieces only 23 configurations are 100-extendable. the 4x5x6 box: (unique and full rotational symmetry of order 4) 31 31 32 18 12 12 32 32 32 18 21 12 33 26 26 18 18 12 33 27 26 19 19 19 33 33 26 19 13 13 31 24 21 18 15 10 32 24 21 21 21 10 33 24 24 17 17 12 30 27 27 27 17 19 30 25 26 27 17 13 31 24 22 23 15 10 31 23 23 23 15 14 28 25 25 15 15 11 29 25 20 20 20 13 30 25 20 16 17 13 28 28 22 14 10 10 28 23 22 14 14 14 28 22 22 16 16 11 29 29 29 16 20 11 30 30 29 16 11 11 the 4x10x14 box: (the only point-symmetric solution) 63 63 64 35 28 28 28 2 2 2 2 2 2 2 63 49 49 49 33 25 28 2 2 2 2 2 2 2 63 50 50 49 33 25 16 2 2 2 2 2 2 2 59 59 60 33 33 15 15 2 2 2 2 2 2 2 59 46 39 39 39 15 2 2 2 2 2 2 2 2 59 46 37 37 39 15 2 2 2 2 2 2 2 2 55 55 56 36 22 20 2 2 2 2 2 2 2 2 55 44 44 36 22 20 12 2 2 2 2 2 2 2 55 42 36 36 20 20 12 2 2 2 2 2 2 2 52 42 42 42 18 12 12 2 2 2 2 2 2 2 64 64 64 35 28 16 16 2 2 2 2 2 2 2 64 49 48 35 33 34 16 2 2 2 2 2 2 2 63 50 48 35 35 25 16 2 2 2 2 2 2 2 60 60 60 40 26 26 10 10 2 2 2 2 2 2 60 46 39 40 24 24 24 2 2 2 2 2 2 2 59 47 47 37 23 15 24 2 2 2 2 2 2 2 56 56 56 36 22 20 2 2 2 2 2 2 2 2 56 45 44 30 21 21 12 2 2 2 2 2 2 2 55 45 52 30 30 30 18 2 2 2 2 2 2 2 52 52 52 42 18 18 18 2 2 2 2 2 2 2 65 65 65 51 29 29 29 2 2 2 2 2 2 2 62 50 65 34 34 34 29 2 2 2 2 2 2 2 61 50 48 34 25 25 17 2 2 2 2 2 2 2 61 61 61 41 26 27 11 10 2 2 2 2 2 2 58 46 46 40 24 14 23 2 2 2 2 2 2 2 57 47 38 37 23 23 23 2 2 2 2 2 2 2 57 57 57 37 22 22 2 2 2 2 2 2 2 2 54 45 44 31 31 21 13 2 2 2 2 2 2 2 53 43 44 31 32 30 13 2 2 2 2 2 2 2 53 53 53 31 19 13 13 2 2 2 2 2 2 2 65 51 51 51 29 17 17 2 2 2 2 2 2 2 62 51 41 41 27 27 17 2 2 2 2 2 2 2 62 48 48 41 26 27 17 2 2 2 2 2 2 2 62 62 61 41 26 27 11 10 2 2 2 2 2 2 58 47 40 40 38 14 11 10 2 2 2 2 2 2 58 47 38 38 38 14 11 11 2 2 2 2 2 2 58 58 57 32 32 14 14 2 2 2 2 2 2 2 54 45 45 43 32 21 13 2 2 2 2 2 2 2 54 43 43 43 32 21 19 2 2 2 2 2 2 2 54 54 53 31 19 19 19 2 2 2 2 2 2 2 the 4x9x15 box: 61 61 43 40 28 28 14 2 2 2 2 2 2 2 2 61 49 43 40 36 21 14 2 2 2 2 2 2 2 2 61 49 40 40 36 21 21 21 2 2 2 2 2 2 2 57 47 47 34 26 26 26 12 2 2 2 2 2 2 2 57 57 57 34 34 34 26 12 2 2 2 2 2 2 2 55 46 46 37 24 24 12 12 2 2 2 2 2 2 2 55 55 55 37 30 16 16 2 2 2 2 2 2 2 2 51 45 37 37 30 16 10 10 2 2 2 2 2 2 2 51 51 51 30 30 16 11 11 2 2 2 2 2 2 2 62 62 43 40 29 28 14 2 2 2 2 2 2 2 2 60 49 42 42 29 29 29 2 2 2 2 2 2 2 2 61 48 48 48 36 22 22 21 2 2 2 2 2 2 2 58 58 47 48 26 20 20 12 2 2 2 2 2 2 2 58 53 57 33 25 34 19 2 2 2 2 2 2 2 2 58 53 46 37 25 24 19 2 2 2 2 2 2 2 2 53 53 55 45 30 17 17 2 2 2 2 2 2 2 2 52 45 45 45 38 18 18 10 2 2 2 2 2 2 2 52 54 51 31 31 16 11 2 2 2 2 2 2 2 2 63 62 43 43 44 28 14 14 2 2 2 2 2 2 2 60 49 49 42 27 28 29 2 2 2 2 2 2 2 2 60 48 41 36 36 22 23 13 2 2 2 2 2 2 2 60 60 47 35 35 35 20 13 2 2 2 2 2 2 2 56 53 47 33 25 35 13 13 2 2 2 2 2 2 2 58 59 46 33 39 24 19 2 2 2 2 2 2 2 2 54 54 46 33 33 24 17 2 2 2 2 2 2 2 2 52 54 38 38 38 18 11 10 2 2 2 2 2 2 2 50 54 38 31 32 15 11 10 2 2 2 2 2 2 2 63 62 44 44 44 23 23 2 2 2 2 2 2 2 2 63 62 44 42 27 22 23 2 2 2 2 2 2 2 2 63 63 41 42 27 22 23 13 2 2 2 2 2 2 2 59 59 41 35 27 27 20 2 2 2 2 2 2 2 2 56 59 41 41 25 25 20 2 2 2 2 2 2 2 2 56 59 39 39 39 19 19 2 2 2 2 2 2 2 2 56 56 39 32 32 18 17 2 2 2 2 2 2 2 2 52 52 50 31 32 18 17 15 2 2 2 2 2 2 2 50 50 50 31 32 15 15 15 2 2 2 2 2 2 2 the 5x7x24 box: 176173157151151136136123115115115 95 95 95 87 67 67 67 61 39 39 39 18 14 176173157152152143136128115109109 95 87 87 87 67 61 61 61 39 31 31 18 14 176176157157152143136137116116100100 88 88 72 72 72 58 47 47 47 24 18 18 177165165165152137137137117116101100 89 81 72 82 62 58 47 50 32 32 32 25 177165158158146146130130117116101100 89 82 82 82 62 50 50 50 32 25 25 25 176173157151144135127123123123115 99 86 95 70 70 59 67 45 45 45 39 18 14 175162162163144143127128114123109108 87 81 81 70 61 66 45 38 31 23 23 15 175163163163144144136128114108108108 88 81 71 70 72 58 46 38 47 24 24 24 175175174165152137128128114114113 94 94 81 71 80 60 60 46 38 38 37 32 24 177164164158145146129130117116101100 89 82 71 71 62 50 46 46 44 25 17 17 173173156151144135135135112107109 99 86 86 86 66 59 59 59 43 45 22 14 14 175162156151143143127135114107109 99 85 85 86 66 66 66 59 38 31 22 23 15 174163156156145145129129113108 99 99 88 80 71 70 58 58 46 37 31 22 22 15 174174174158145146129130113113113 94 88 80 80 80 68 60 44 37 37 37 15 15 177177164158145146129130117117101101 89 89 69 69 62 62 44 44 44 36 16 17 170170156148142142125125112112112 92 92 92 74 65 55 55 56 43 29 22 23 10 171162160148147142127127126107112 92 85 78 74 65 56 56 56 43 29 33 23 10 171162160149149142126126126105105 93 93 84 68 65 65 60 43 43 29 29 16 16 171171164150149133133134119106105 94 93 84 68 68 68 60 57 36 30 30 16 17 172172164150149134134134119106105 94 93 79 79 69 57 57 57 36 36 36 16 17 170166154154141141125132111111111 91 91 92 74 65 55 51 42 33 29 20 20 10 171166160148147132132132111107107103 85 78 78 78 56 51 42 33 33 33 20 11 168161161161147142126121121121 98 98 85 84 73 78 53 53 54 34 27 27 20 21 169161155147147133131121119104104 98 97 77 73 69 54 54 54 35 30 21 21 21 169172155150149134131122122106105 98 93 77 79 69 57 49 49 35 28 28 13 13 170166154153141138125118110103111 91 83 74 74 75 55 51 40 34 34 19 10 10 170160160148148132125118124103103103 90 75 75 75 55 63 42 34 26 26 26 11 168168168161140140140118118121 97102 84 84 73 64 53 52 41 34 26 27 20 11 167172168159140133131119119104 97 97 97 76 79 64 54 48 48 48 30 21 11 11 169172155150150133139120122106106 98 96 77 79 64 64 48 49 35 30 28 12 13 166166154153141138124118110110110 91 83 83 83 63 51 51 40 40 40 19 19 19 167167154153141138124124124102110 91 90 75 83 63 63 63 42 42 40 27 26 19 167159153153138138140120120102102102 90 73 73 64 53 52 41 41 41 27 12 12 167159159159139131131120122104 96 90 90 76 76 76 53 52 49 48 41 28 12 13 169169155155139139139120122104 96 96 96 77 77 76 52 52 49 35 35 28 12 13 the 5x7x30 box: point symmetric solution possible 113100100 88 88 88 65 65 65 51 38 38 38 20 15 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 113101100 88 87 75 65 63 55 51 38 44 30 20 15 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 114101100 89 89 75 66 57 57 44 44 44 30 20 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 114101101 90 89 75 75 76 57 45 45 46 25 25 25 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 114114112 90 89 76 76 76 57 46 46 46 25 21 21 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 111111 95 95 95 88 66 66 65 51 35 35 38 20 15 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 113101 95 92 87 75 66 63 55 55 55 35 30 18 18 18 2 2 2 2 2 2 2 2 2 2 2 2 2 2 114 98100 92 87 78 66 63 54 44 55 35 31 18 10 10 2 2 2 2 2 2 2 2 2 2 2 2 2 2 112 99 99 87 87 78 63 63 62 45 36 36 37 19 25 10 2 2 2 2 2 2 2 2 2 2 2 2 2 2 112112112 90 89 76 64 64 57 46 37 37 37 19 21 10 2 2 2 2 2 2 2 2 2 2 2 2 2 2 111108 97 86 95 73 73 73 51 51 34 34 34 15 15 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 113113 97 92 80 73 71 56 52 52 34 30 30 16 16 18 2 2 2 2 2 2 2 2 2 2 2 2 2 2 109 98 98 98 80 78 62 56 54 54 54 35 31 17 17 17 2 2 2 2 2 2 2 2 2 2 2 2 2 2 109 99110 98 80 80 62 62 62 45 54 36 31 17 21 14 2 2 2 2 2 2 2 2 2 2 2 2 2 2 110110110 90 90 74 74 64 53 45 37 31 31 19 21 10 2 2 2 2 2 2 2 2 2 2 2 2 2 2 111108 97 86 86 86 61 73 50 50 50 32 34 13 13 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 111108 92 92 80 86 71 70 50 52 40 32 24 16 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 108108 93 93 78 78 71 56 53 53 40 36 24 14 2 17 2 2 2 2 2 2 2 2 2 2 2 2 2 2 109 99 94 93 79 71 71 64 53 42 42 36 33 14 14 14 2 2 2 2 2 2 2 2 2 2 2 2 2 2 110 99 94 93 79 72 74 64 53 43 33 33 33 19 19 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 105 97 97 85 81 70 61 61 61 52 50 32 23 13 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 105103106 85 81 70 70 70 61 52 40 28 23 16 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 106106106 85 85 72 72 56 56 43 43 28 24 16 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 109109 91 91 77 72 74 59 49 43 42 28 28 26 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 107107 94 93 79 72 74 59 49 43 33 29 29 12 12 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 102 96 96 85 81 68 60 60 60 39 32 32 22 13 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 105103 96 83 83 83 60 69 47 40 40 28 23 13 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 106103 96 83 82 69 69 69 58 41 42 26 24 24 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 103103 91 84 77 77 77 59 48 41 42 26 26 26 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 104107 94 94 79 79 77 67 49 41 41 27 29 11 12 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 102102102 81 81 68 68 68 60 39 39 39 22 22 22 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 105105102 84 84 83 58 68 47 47 47 39 23 23 22 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 104104 96 84 82 69 58 58 58 41 47 27 27 11 11 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 104107 91 84 82 67 59 59 48 48 48 27 29 11 12 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 104107 91 82 82 67 67 67 49 49 48 27 29 11 12 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 References: [1] H. Postl many new prime packings. letter from 9. July 1998