Prime Packings of the Pseudo-Tetromino j: Torsten Sillke, FRA initial version: 1993 Update 1997-11-13: 14x36 H. Postl Update 1998-07-02: 14x48, 6x6x6 Update 1998-07-06: 4x5x9, 3|volume Update 1998-07-08: 18xN strip Update 1998-07-10: 5x6x6, 18x{36,64,68}, 18x{48,76,80} Update 1998-07-11: 18x52, 18x56 Update 1998-07-12: 18x60, 14x60, 6x6x7 Update 1998-07-13: 20x{30,36,42,48,52}, no 24x24 Update 1998-07-09:[3] 5x5x12 H. Postl Update 1998-07-09:[3] 19x72; 20x30,36,42,48,54 H. Postl Update 1998-07-09:[3] 21x32,40,48,56; 22x48,60,72 H. Postl 3 dim complete Update 1999-02-02:[4] 19x48, 22x36, 23x48 M. Reid Update 1999-02-02:[4] width <= 22 complete M. Reid Open case: 3x3xZ x x x x . = j4 2-dim: ------ Nx 14, 16p, 18, 19, 20, 21, 22 14x 36p, 48p, 60p, ... {36, 48, 60}+36n 18x 36p, 48p, 52p, 56p, 60p, 64p, 68p, 72, 76p, 80p, ... {48,52,56,60,64,68,72,76,80}+36n 19x 48p, 72p ... {48, 72}+48n 20x 30p, 36p, 42p, 48p, 54p, ... {30,36,42,48,54}+30n 21x 32p, 40p, 48, 56, ... {32, 40, 48, 56}+32n 22x 36p, 48p, 60p, ... {36,48,60}+36n 23x 48p 3-dim: ------ 3x4x 4p, 6p, 7p, 8, 9p, ... {6..9}+4n 3x6x 4p, 8, 10p, ... {8, 10}+4n 3x8x 4, 5p, 6, 7, ... {4..7}+4n 3x10x 4, 6, ... {4, 6}+4n 3x12x 5p, 6, 7, ... {4..7}+4n 4x4x 3p, ... 3n 5x4x 6p, 9p, ... {6, 9}+6n 6x4x 3p, 4, 5p, ... {3..5}+3n 6x5x 4p, 6p, ... {4, 6}+4n 6x6x 4, 5p, 6p, 7p, ... {4..7}+4n 5x5x 12p, ... 12n Complete list of prime boxes (# = 13): 3x4x{4,6,7,9}, 3x5x{8,12}, 3x6x10, 4x5x{6,9}, 5x5x12, 5x6x6, 6x6x{6,7}. Impossible: Nx{3..13,15,17} 16xk (no rectangle of width 16, stripes only) 18x{20, 24, 28, 32, 40, 44} (Begin and End part see 18*N) 24x{14,18,19,20,24} 3xNx{3} 3x4x{3,5} 3x6x{6} Thm[0]: no odd rectangle possible Proof: color the square (x,y) with x (modulo 2). The v piece covers an odd number of squares of each color. As both colors appear an even number of times the number of pieces must be even. Thm[2,3]: 3 | Volume Proof: color the cube (x,y,z) with x+y+z (modulo 3). The j4 piece covers an even number of cubes of each color. Therefore the parity of the three colors in the box must be the same. Annotations: circle symmetry: x x x . . y x . . y y y the 14*36 rectangle (2 solutions (without circle symmetry)) These solutions are symmetric (ignoring circle symmetry). solution 1: A B1 circle B1' A' solution 2: A B2 circle B2' A' The prime indicates a reflected part. part A: 14*10 without two dominoes (unique) 36 36 32 32 26 26 16 16 10 10 37 32 36 26 22 27 17 10 16 . 37 32 36 26 22 27 17 10 16 . 38 37 37 27 27 22 22 17 17 12 38 39 39 33 23 28 18 11 11 12 39 38 38 33 23 28 18 12 12 11 39 33 33 28 28 23 23 18 18 11 40 40 34 34 29 29 19 19 13 13 41 34 40 29 24 30 20 13 19 . 41 34 40 29 24 30 20 13 19 . 42 41 41 30 30 24 24 20 20 15 42 43 43 35 25 31 21 14 14 15 43 42 42 35 25 31 21 15 15 14 43 35 35 31 31 25 25 21 21 14 part B1: with 3 circles . 35 35 24 24 15 15 C+ C+ 35 29 29 27 27 24 C+ 15 . C+ 35 30 27 29 19 24 C+ 15 . C+ . 30 27 29 19 D+ D+ C+ C+ . 31 30 30 D+ 19 19 D+ . 11 . 31 32 32 D+ 21 21 D+ . 11 . 32 31 31 21 D+ D+ 11 11 . 32 37 25 21 17 17 12 12 36 36 37 25 26 26 22 17 . 12 37 37 36 26 25 25 22 17 . 12 . 33 36 26 22 22 E+ E+ - - . 33 34 34 28 E+ 23 - E+ . - . 34 33 33 28 E+ 23 - E+ . - . 34 28 28 23 23 E+ E+ - - part B2: with 2 circles . 35 35 24 24 15 15 C+ C+ 35 29 29 27 27 24 C+ 15 . C+ 35 30 27 29 19 24 C+ 15 . C+ . 30 27 29 19 20 20 C+ C+ . 31 30 30 20 19 19 11 11 . 31 32 32 20 21 21 17 . 11 . 32 31 31 21 16 16 17 . 11 . 32 37 25 21 17 17 16 . 12 36 36 37 25 26 26 22 16 . 12 37 37 36 26 25 25 22 12 12 . 33 36 26 22 22 E+ E+ - - . 33 34 34 28 E+ 23 - E+ . - . 34 33 33 28 E+ 23 - E+ . - . 34 28 28 23 23 E+ E+ - - the 14*48 rectangle part B: (solution: A B A') . 07 07 96 96 87 87 81 81 75 75 71 71 63 63 52 52 46 46 39 39 33 33 29 29 21 21 10 10 . 07 01 01 99 99 96 85 87 77 81 71 75 63 61 61 57 57 52 44 46 35 39 29 33 21 19 19 13 13 10 07 02 99 01 91 96 85 87 77 81 71 75 63 67 57 61 58 52 44 46 35 39 29 33 21 25 13 19 14 10 . 02 99 01 91 88 88 85 85 77 77 66 66 67 57 61 58 47 47 44 44 35 35 24 24 25 13 19 14 . . 03 02 02 92 91 91 88 78 82 72 67 67 66 58 58 53 53 49 47 36 40 30 25 25 24 14 14 16 . . 03 04 04 92 93 93 88 78 82 72 68 68 66 59 53 48 48 49 47 36 40 30 26 26 24 15 15 16 . . 04 03 03 93 92 92 82 82 78 78 72 72 68 59 53 49 49 48 40 40 36 36 30 30 26 16 16 15 . . 04 09 97 93 89 89 83 83 79 79 73 73 68 65 59 59 55 48 45 45 37 37 31 31 26 23 11 15 . 08 08 09 97 98 98 94 89 79 83 73 69 64 64 65 54 54 55 45 41 37 34 31 27 22 22 23 11 12 12 09 09 08 98 97 97 94 89 79 83 73 69 65 65 64 55 55 54 45 41 37 34 31 27 23 23 22 12 11 11 . 05 08 98 94 94 86 86 80 80 74 74 69 69 64 60 60 54 51 42 41 41 34 34 27 27 22 12 18 . . 05 06 06 00 90 95 84 86 76 80 70 74 62 60 56 50 50 51 42 43 43 38 28 32 20 17 17 18 . . 06 05 05 00 90 95 84 86 76 80 70 74 62 60 56 51 51 50 43 42 42 38 28 32 20 18 18 17 . . 06 00 00 95 95 90 90 84 84 76 76 70 70 62 62 56 56 50 43 38 38 32 32 28 28 20 20 17 . the 14*60 rectangle part F: (solution: A F A') .143141141133133127127119119117117108108106 96 91 91 85 85 77 77 75 75 66 66 64 54 49 49 43 43 35 35 29 29 24 24 19 19 14 . .143144144141129133123127117119108105105106 96 97 97 91 81 85 75 77 66 63 63 64 54 55 55 49 39 43 33 35 24 29 19 13 13 14 . .144143143141129133123127117119108106106105 97 96 96 91 81 85 75 77 66 64 64 63 55 54 54 49 39 43 33 35 24 29 19 14 14 13 . .144150138134134129129123123113113109109105 97 92 92 87 87 81 81 71 71 67 67 63 55 50 50 45 45 39 39 33 33 25 25 22 10 13 . 149149150138139139134130120124114109113102102 99 99 87 92 88 78 82 72 67 71 60 60 57 57 45 50 46 36 40 30 25 21 21 22 10 11 11 150150149139138138134130120124114109113110 99102 93 87 92 88 78 82 72 67 71 68 57 60 51 45 50 46 36 40 30 25 22 22 21 11 10 10 .145149139135130130124124120120114114110 99102 93 88 88 82 82 78 78 72 72 68 57 60 51 46 46 40 40 36 36 30 30 27 21 11 16 . .145146146135131131125125118118110110107107104 94 93 93 83 83 76 76 68 68 65 65 62 52 51 51 41 41 37 37 26 26 27 15 15 16 . .146145145136135135131121125115118107103103104 94 95 95 89 79 83 73 76 65 61 61 62 52 53 53 47 37 41 31 27 27 26 16 16 15 . .146147147136137137131121125115118107104104103 95 94 94 89 79 83 73 76 65 62 62 61 53 52 52 47 37 41 31 28 28 26 17 17 15 . .147142148137136136126126121121115115112100103 95 89 89 84 84 79 79 73 73 70 58 61 53 47 47 42 42 34 34 31 31 28 18 20 17 . 151147142148137140128132122126116111111112100101101 98 86 90 80 84 74 69 69 70 58 59 59 56 44 48 38 42 32 34 23 28 18 20 17 12 151148148142142140128132122126116112112111101100100 98 86 90 80 84 74 70 70 69 59 58 58 56 44 48 38 42 32 34 23 20 20 18 18 12 .151151140140132132128128122122116116111101 98 98 90 90 86 86 80 80 74 74 69 59 56 56 48 48 44 44 38 38 32 32 23 23 12 12 . the 16*N strip: (unique (without circle symmetry)) 91 91 86 86 80 80 68 68 62 62 56 56 the only extendable border 92 86 91 80 74 81 69 62 68 56 50 57 92 86 91 80 74 81 69 62 68 56 50 57 93 92 92 81 81 74 74 69 69 57 57 50 50 93 C2 C2 82 82 75 75 63 63 58 58 51 51 C2 93 93 C2 75 82 76 64 70 63 51 58 52 C2 C1 C1 C2 75 82 76 64 70 63 51 58 52 C1 C2 C2 C1 76 76 70 70 64 64 52 52 C1 C3 C3 C1 77 77 71 71 65 65 53 53 C3 C1 C1 C3 78 83 77 65 71 66 54 59 53 C3 97 97 C3 78 83 77 65 71 66 54 59 53 97 C3 C3 83 83 78 78 66 66 59 59 54 54 97 98 98 84 84 79 79 72 72 60 60 55 55 98 90 99 85 79 84 72 67 73 61 55 60 98 90 99 85 79 84 72 67 73 61 55 60 99 99 90 90 85 85 73 73 67 67 61 61 the 18*N strip: front part C: this is the only infinitly extendable figure (n=42) (unique) 42 42 38 38 28 28 14 14 10 10 43 38 42 28 25 29 15 10 14 . 43 38 42 28 25 29 15 10 14 . 44 43 43 29 29 25 25 15 15 . 44 45 45 30 30 23 23 16 16 . 45 44 44 31 36 30 16 23 17 . 45 46 46 31 36 30 16 23 17 . 46 39 36 36 31 31 17 17 19 . 46 39 40 40 37 24 18 18 19 . 47 40 39 39 37 24 19 19 18 11 47 40 37 37 32 32 24 24 18 11 48 47 47 32 26 33 20 11 11 . 48 49 49 32 26 33 20 12 12 . 49 48 48 33 33 26 26 20 20 12 49 50 50 34 34 27 27 21 21 12 50 41 51 35 27 34 21 13 22 . 50 41 51 35 27 34 21 13 22 . 51 51 41 41 35 35 22 22 13 13 part of length 28 is unique (upto circle symmetry): . 29 29 14 14 10 10 02 02 96 84 79 79 69 69 62 62 51 51 C+ C+ 38 38 28 28 14 14 10 10 29 23 30 15 10 14 02 95 95 96 84 85 85 79 74 69 66 62 C+ 51 38 C+ 28 25 29 15 10 14 . 29 23 30 15 10 14 02 96 96 95 85 84 84 79 74 69 66 62 C+ 51 38 C+ 28 25 29 15 10 14 . 30 30 23 23 15 15 03 97 97 95 85 80 74 74 66 66 63 63 57 C+ C+ 44 29 29 25 25 15 15 . 31 31 24 24 16 16 03 04 04 97 92 80 81 81 75 63 56 56 57 43 43 44 30 30 23 23 16 16 . 32 24 31 16 11 17 04 03 03 97 92 81 80 80 75 63 57 57 56 44 44 43 31 36 30 16 23 17 . 32 24 31 16 11 17 04 08 92 92 86 81 75 75 70 70 64 64 56 53 39 43 31 36 30 16 23 17 . 33 32 32 17 17 11 11 08 98 98 86 87 87 70 76 64 58 52 52 53 39 36 36 31 31 17 17 19 . 33 25 25 18 18 08 08 98 93 99 87 86 86 70 76 64 58 53 53 52 49 39 39 37 24 18 18 19 . . 33 33 25 21 18 05 98 93 99 87 90 76 76 67 65 65 58 58 52 49 40 40 37 24 19 19 18 11 . 34 34 25 21 18 05 99 99 93 93 90 82 82 67 68 68 65 49 49 40 37 37 32 32 24 24 18 11 34 26 21 21 12 09 09 05 05 90 90 82 77 83 68 67 67 65 54 54 40 46 32 26 33 20 11 11 . 34 26 27 27 12 13 13 09 94 00 88 82 77 83 68 71 59 54 50 45 45 46 32 26 33 20 12 12 . . 27 26 26 13 12 12 09 94 00 88 83 83 77 77 71 59 54 50 46 46 45 33 33 26 26 20 20 12 . 27 28 28 13 20 06 00 00 94 94 88 88 71 71 73 60 59 59 50 50 45 34 34 27 27 21 21 12 35 28 22 19 19 20 06 07 07 01 89 91 78 72 72 73 60 61 61 55 41 47 35 27 34 21 13 22 . 35 28 22 20 20 19 07 06 06 01 89 91 78 73 73 72 61 60 60 55 41 47 35 27 34 21 13 22 . . 35 35 22 22 19 07 01 01 91 91 89 89 78 78 72 61 55 55 47 47 41 41 35 35 22 22 13 13 part of length 32: .203203189189183183177177172159153153145145140140127127121121113113 97 97 94 94 84 84 70 70 66 66 203197204190183189177171171172159160160153140145141128121127113109114 98 94 97 84 81 85 71 66 70 . 203197204190183189177172172171160159159153140145141128121127113109114 98 94 97 84 81 85 71 66 70 . 204204197197190190178173173171160154147147141141133133128128114114109109 98 98 85 85 81 81 71 71 . 205205198198191191178179179173165154155155147133131131122122117117104 99 99 92 92 86 86 79 79 73 . 206198205191184192179178178173165155154154147133132134131117122118104105105 99 86 92 79 72 72 73 . 206198205191184192179180165165167155156156146146132134131117122118105104104 99 86 92 79 73 73 72 . 207206206192192184184180166166167156148146142134134132132118118110105100100 93 93 87 87 74 74 72 . 207199199193193180180174167167166156148146142135135129129123123110111111106100 87 93 88 75 80 74 . .207207199195193185174175175166161161148148142142135123129115111110110106100 87 93 88 75 80 74 67 .208208199195193185175174174161157157149149136136135123129115111106106 95 95 88 88 80 80 75 75 67 208200195195185185187175176176161163149157150137143136124119119115115107107102 95 82 89 76 67 67 . 208200201201186186187176168162162163149157150137143136124125125119107101101102 95 82 89 76 68 68 . .201200200187187186176168163163162150150143143137137125124124119107102102101 89 89 82 82 76 76 68 .201202202194194186182169168168162151151144144138138125120120112112103103101 90 90 83 83 77 77 68 209202196194188181181182169170170164152158151138144139126130116120108112 96103 91 83 90 77 69 78 . 209202196194188182182181170169169164152158151138144139126130116120108112 96103 91 83 90 77 69 78 . .209209196196188188181170164164158158152152139139130130126126116116108108 96 96 91 91 78 78 69 69 Note: the only other first part which is extendable at lease 100 pieces is the following. But it dies out after 181 pieces. * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * . * * * * * * * * . * * * * * * * * . * * * * * * * * * * * * * * * * * * * * * * * * * * * * . * * * * * * * * . * * * * * * * * . * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * the 18*(36+28n+32m) rectangle: (2 solutions, both symmetric) the 18*(48+28n+32m) rectangle: the 18*(60+28n+32m) rectangle: . 42 42 28 28 22 22 16 16 11 2 2 2 2 2 2 2 . part D1 42 36 43 29 22 28 16 10 10 11 2 2 2 2 2 2 2 2 42 36 43 29 22 28 16 11 11 10 2 2 2 2 2 2 2 2 C & D1 & D1 & C 43 43 36 36 29 29 17 12 12 10 2 2 2 2 2 2 2 . is 18x36 44 44 37 37 30 30 17 18 18 12 2 2 2 2 2 2 2 . 45 37 44 30 23 31 18 17 17 12 2 2 2 2 2 2 2 2 45 37 44 30 23 31 18 19 2 2 2 2 2 2 2 2 2 2 46 45 45 31 31 23 23 19 2 2 2 2 2 2 2 2 2 . 46 38 38 32 32 19 19 13 2 2 2 2 2 2 2 2 2 . . 46 46 38 34 32 24 13 14 14 2 2 2 2 2 2 2 2 . 47 47 38 34 32 24 14 13 13 2 2 2 2 2 2 2 2 47 39 34 34 24 24 26 14 15 15 2 2 2 2 2 2 2 2 47 39 40 40 25 25 26 15 2 2 2 2 2 2 2 2 2 2 . 40 39 39 26 26 25 15 2 2 2 2 2 2 2 2 2 2 . 40 41 41 33 33 25 21 2 2 2 2 2 2 2 2 2 2 48 41 35 33 27 20 20 21 2 2 2 2 2 2 2 2 2 2 48 41 35 33 27 21 21 20 2 2 2 2 2 2 2 2 2 2 . 48 48 35 35 27 27 20 2 2 2 2 2 2 2 2 2 . . 42 42 28 28 22 22 16 16 11 2 2 2 2 2 2 2 . part D2 42 36 43 29 22 28 16 10 10 11 2 2 2 2 2 2 2 2 42 36 43 29 22 28 16 11 11 10 2 2 2 2 2 2 2 2 C & D2 & D2 & C 43 43 36 36 29 29 17 12 12 10 2 2 2 2 2 2 2 . is 18x36 44 44 37 37 30 30 17 18 18 12 2 2 2 2 2 2 2 . 45 37 44 30 23 31 18 17 17 12 2 2 2 2 2 2 2 2 45 37 44 30 23 31 18 19 2 2 2 2 2 2 2 2 2 2 46 45 45 31 31 23 23 19 13 13 2 2 2 2 2 2 2 . 46 38 38 32 32 19 19 13 2 2 2 2 2 2 2 2 2 . . 46 46 38 34 32 24 13 14 14 2 2 2 2 2 2 2 2 . 47 47 38 34 32 24 14 2 2 2 2 2 2 2 2 2 2 47 39 34 34 24 24 26 14 15 15 2 2 2 2 2 2 2 2 47 39 40 40 25 25 26 15 2 2 2 2 2 2 2 2 2 2 . 40 39 39 26 26 25 15 2 2 2 2 2 2 2 2 2 2 . 40 41 41 33 33 25 21 2 2 2 2 2 2 2 2 2 2 48 41 35 33 27 20 20 21 2 2 2 2 2 2 2 2 2 2 48 41 35 33 27 21 21 20 2 2 2 2 2 2 2 2 2 2 . 48 48 35 35 27 27 20 2 2 2 2 2 2 2 2 2 . . 96 96 81 81 77 77 68 68 55 55 50 50 41 41 28 28 22 22 16 16 11 2 2 2 2 2 part E1 96 90 97 82 77 81 68 63 69 56 50 55 41 37 42 29 22 28 16 10 10 11 2 2 2 2 2 96 90 97 82 77 81 68 63 69 56 50 55 41 37 42 29 22 28 16 11 11 10 2 2 2 2 2 C & D1 & E1 & C 97 97 90 90 82 82 69 69 63 63 56 56 42 42 37 37 29 29 17 12 12 10 2 2 2 2 2 is 18x48 98 98 91 91 83 83 70 70 64 64 57 57 43 43 38 38 30 30 17 18 18 12 2 2 2 2 2 99 91 98 83 78 84 71 64 70 57 51 58 44 38 43 30 23 31 18 17 17 12 2 2 2 2 2 C & E1 & E1 & C 99 91 98 83 78 84 71 64 70 57 51 58 44 38 43 30 23 31 18 19 2 2 2 2 2 2 2 is 18x60 00 99 99 84 84 78 78 71 71 58 58 51 51 44 44 31 31 23 23 19 2 2 2 2 2 2 2 00 92 92 85 85 75 75 65 65 61 61 52 52 48 34 32 32 19 19 13 2 2 2 2 2 2 2 . 00 00 92 88 85 72 75 61 65 52 47 47 48 34 35 35 32 24 13 14 14 2 2 2 2 2 . 01 01 92 88 85 72 75 61 65 52 48 48 47 35 34 34 32 24 14 13 13 2 2 2 2 2 01 93 88 88 79 76 76 72 72 59 59 53 53 47 35 40 24 24 26 14 15 15 2 2 2 2 2 01 93 94 94 79 80 80 76 62 66 53 59 45 39 39 40 25 25 26 15 2 2 2 2 2 2 2 . 94 93 93 80 79 79 76 62 66 53 59 45 40 40 39 26 26 25 15 2 2 2 2 2 2 2 . 94 95 95 80 87 73 66 66 62 62 49 49 45 45 39 33 33 25 21 2 2 2 2 2 2 2 02 95 89 86 86 87 73 74 74 67 54 60 46 49 36 33 27 20 20 21 2 2 2 2 2 2 2 02 95 89 87 87 86 74 73 73 67 54 60 46 49 36 33 27 21 21 20 2 2 2 2 2 2 2 . 02 02 89 89 86 74 67 67 60 60 54 54 46 46 36 36 27 27 20 2 2 2 2 2 2 2 . 96 96 81 81 77 77 68 68 55 55 50 50 41 41 28 28 22 22 16 16 11 2 2 2 2 2 part E2 96 90 97 82 77 81 68 63 69 56 50 55 41 37 42 29 22 28 16 10 10 11 2 2 2 2 2 96 90 97 82 77 81 68 63 69 56 50 55 41 37 42 29 22 28 16 11 11 10 2 2 2 2 2 C & D2 & E2 & C 97 97 90 90 82 82 69 69 63 63 56 56 42 42 37 37 29 29 17 12 12 10 2 2 2 2 2 is 18x48 98 98 91 91 83 83 70 70 64 64 57 57 43 43 38 38 30 30 17 18 18 12 2 2 2 2 2 99 91 98 83 78 84 71 64 70 57 51 58 44 38 43 30 23 31 18 17 17 12 2 2 2 2 2 C & E2 & E2 & C 99 91 98 83 78 84 71 64 70 57 51 58 44 38 43 30 23 31 18 19 2 2 2 2 2 2 2 is 18x60 00 99 99 84 84 78 78 71 71 58 58 51 51 44 44 31 31 23 23 19 13 13 2 2 2 2 2 00 92 92 85 85 75 75 65 65 61 61 52 52 48 34 32 32 19 19 13 2 2 2 2 2 2 2 . 00 00 92 88 85 72 75 61 65 52 47 47 48 34 35 35 32 24 13 14 14 2 2 2 2 2 . 01 01 92 88 85 72 75 61 65 52 48 48 47 35 34 34 32 24 14 2 2 2 2 2 2 2 01 93 88 88 79 76 76 72 72 59 59 53 53 47 35 40 24 24 26 14 15 15 2 2 2 2 2 01 93 94 94 79 80 80 76 62 66 53 59 45 39 39 40 25 25 26 15 2 2 2 2 2 2 2 . 94 93 93 80 79 79 76 62 66 53 59 45 40 40 39 26 26 25 15 2 2 2 2 2 2 2 . 94 95 95 80 87 73 66 66 62 62 49 49 45 45 39 33 33 25 21 2 2 2 2 2 2 2 02 95 89 86 86 87 73 74 74 67 54 60 46 49 36 33 27 20 20 21 2 2 2 2 2 2 2 02 95 89 87 87 86 74 73 73 67 54 60 46 49 36 33 27 21 21 20 2 2 2 2 2 2 2 . 02 02 89 89 86 74 67 67 60 60 54 54 46 46 36 36 27 27 20 2 2 2 2 2 2 2 the 18*(56+28n+32m) rectangle: (unique solution, the solution is point symmetric) . 87 87 72 72 68 68 60 60 54 42 37 37 27 27 19 19 2 2 2 2 2 2 2 2 2 2 87 81 88 73 68 72 60 53 53 54 42 43 43 37 32 27 24 19 2 2 2 2 2 2 2 2 2 87 81 88 73 68 72 60 54 54 53 43 42 42 37 32 27 24 19 2 2 2 2 2 2 2 2 2 88 88 81 81 73 73 61 55 55 53 43 38 32 32 24 24 20 20 2 2 2 2 2 2 2 2 2 89 89 82 82 74 74 61 62 62 55 50 38 39 39 33 20 14 14 2 2 2 2 2 2 2 2 2 90 82 89 74 69 75 62 61 61 55 50 39 38 38 33 20 15 21 14 2 2 2 2 2 2 2 2 90 82 89 74 69 75 62 66 50 50 44 39 33 33 28 28 15 21 14 11 2 2 2 2 2 2 2 91 90 90 75 75 69 69 66 56 56 44 45 45 28 34 21 21 15 15 11 2 2 2 2 2 2 2 91 83 83 76 76 66 66 56 51 57 45 44 44 28 34 25 25 11 11 2 2 2 2 2 2 2 2 . 91 91 83 79 76 63 56 51 57 45 48 34 34 25 22 22 16 16 2 2 2 2 2 2 2 2 . 92 92 83 79 76 63 57 57 51 51 48 40 40 25 30 16 22 2 2 2 2 2 2 2 2 2 92 84 79 79 70 67 67 63 63 48 48 40 35 29 29 30 16 22 2 2 2 2 2 2 2 2 2 92 84 85 85 70 71 71 67 52 58 46 40 35 30 30 29 17 12 12 2 2 2 2 2 2 2 2 . 85 84 84 71 70 70 67 52 58 46 41 41 35 35 29 17 18 18 12 2 2 2 2 2 2 2 . 85 86 86 71 78 64 58 58 52 52 46 46 41 26 31 18 17 17 12 2 2 2 2 2 2 2 93 86 80 77 77 78 64 65 65 59 47 49 36 41 26 31 18 23 10 13 2 2 2 2 2 2 2 93 86 80 78 78 77 65 64 64 59 47 49 36 31 31 26 26 23 10 13 2 2 2 2 2 2 2 . 93 93 80 80 77 65 59 59 49 49 47 47 36 36 23 23 13 13 10 10 2 2 2 2 2 2 the 20*(30+6n) rectangle: (unique point symmetric solution 20*30) 74 74 67 67 61 61 52 52 39 39 36 21 13 13 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 75 67 74 61 54 52 45 39 35 35 36 21 22 22 13 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 75 67 74 61 54 52 45 39 36 36 35 22 21 21 13 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 76 75 75 62 62 54 54 45 45 41 35 22 23 23 15 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 76 77 77 68 55 62 46 40 40 41 26 23 14 14 15 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 77 76 76 68 55 62 46 41 41 40 26 23 15 15 14 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 77 68 68 70 56 55 55 46 46 40 38 26 26 10 14 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 78 69 69 70 56 57 57 53 37 37 38 24 24 10 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 78 70 70 69 57 56 56 53 38 38 37 25 27 24 10 10 2 2 2 2 2 2 2 2 2 2 2 2 2 2 79 78 78 69 57 53 53 42 47 30 37 25 27 24 16 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 79 C+ C+ 63 63 58 58 42 47 30 27 27 25 25 16 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 C+ 79 79 C+ 58 63 47 47 42 42 30 30 16 16 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 C+ D+ D+ C+ 58 63 48 48 43 43 31 31 17 17 11 11 2 2 2 2 2 2 2 2 2 2 2 2 2 2 D+ C+ C+ D+ 59 64 49 43 48 31 28 32 18 11 17 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 D+ 82 82 D+ 59 64 49 43 48 31 28 32 18 11 17 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 82 D+ D+ 64 64 59 59 49 49 32 32 28 28 18 18 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 82 83 83 65 65 60 60 50 50 33 33 29 29 19 19 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 83 73 84 66 60 65 50 44 51 34 29 33 19 12 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 83 73 84 66 60 65 50 44 51 34 29 33 19 12 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 84 84 73 73 66 66 51 51 44 44 34 34 20 20 12 12 2 2 2 2 2 2 2 2 2 2 2 2 2 2 39 39 36 36 21 21 . . . 36 39 35 20 24 21 . . 36 39 35 20 24 21 . . 35 35 24 24 20 20 . . 34 34 23 23 19 19 . . 33 38 34 19 23 18 . . 33 38 34 19 23 18 . 38 38 33 33 18 18 . . . . 32 32 22 22 12 12 . 32 37 22 27 12 17 . . 32 37 22 27 12 17 . 37 37 27 27 17 17 . . . . 31 31 16 16 11 11 . 31 26 30 15 11 16 . . 31 26 30 15 11 16 . . 30 30 26 26 15 15 . . 29 29 25 25 14 14 . . 28 25 29 14 10 13 . . 28 25 29 14 10 13 . . . 28 28 13 13 10 10 the 6x6x6 cube: 6 6 6 6 6 6 6 . = A3 6 6 6 6 6 . = A4 A3 & A3 is 6x6x6 6 6 6 6 6 6 6 6 6 6 6 6 A3 & A4 is 6x6x7 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 94 85 82 82 76 76 95 85 83 83 77 77 96 86 85 85 73 74 97 86 87 87 73 75 98 87 86 86 80 80 99 87 84 84 81 81 94 88 88 76 82 78 95 89 89 77 83 78 96 90 90 78 78 74 97 91 91 79 79 75 98 92 92 80 73 79 99 93 93 81 84 79 88 94 94 76 82 . 89 95 95 77 83 . 90 96 96 74 74 . 91 97 97 75 75 . 92 98 98 80 73 . 93 99 99 81 84 . 88 . . . . . 89 . . . . . 90 . . . . . 91 . . . . . 92 . . . . . 93 . . . . . References: [0] D. A. Klarner, Packing a Rectangle with Congruent N-ominoes. JoCT 7 (1969) 107-115; Zbl 174.41 [1] Helmut Postl, A 14x36 rectangle. postcard from 1997-11-13. [2] Helmut Postl, Pentacube 35 is not boxable. The 5*5*n boxes of the Q pseudo pentomino. letter from 1998-06-08 [3] Helmut Postl, many new prime boxes. letter from 1998-07-09 [4] Michael Reid, prime packing list check email from 1999-02-02