prime boxes of the t-pentomino: Torsten Sillke, BI, 18.04.93 with data from Mike Beeler and Helmut Postl. last update: 20.04.93: 3x11x25no, 3x11x30p, 3x12x25p last update: 21.04.93: 3x13x15no, 3x11x35p, 3x14x15p last update: 22.04.93: 3x11x40p, last update: 25.04.93: 3x15x18p, 3x11x45p, 3x11x50p, 3x16x20p last update: 26.04.93: 3x13x30p last update: 09.07.98: 4x14x20,25,30; no 4x10x11..15; no 5x8x11,12 H Postl [2] last update: 01.07.09: 3x15x15no, 3x15x17p, 3x15x19p, 3x20x13p, 3x20x18p [7] 2-dim: ------ ZxZ 3-dim: ------ 3x10x 10p, 14p, 20, 24, 26p, 27p, 28, 30, 31p, 32p, 33p, 34, 35p, 36, 37, 38, 39p, ... {30..39} + 10n 3x15x 8p, 12p, 14p, 16, 17p, 18p, 19p, 20, 21?, 22, 23?, ... {16..23} + 8n 3x20x 7p, 10, 12p, 13p, 14, 15, 16p, 17, ... {12..17} + 7n 3x25x 12p, 13?, 14, 15, 16?, 17?, 18?, 19?, 20, 21?, 22, 23?, 24, 25?, 26, 27, 3x30x 8, 10, 11p, 12, 13p, 14, 15, 16, 17, ... {10..17} + 8n 3x35x 8p, 10p, 11p, 12, 13?, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25,.. 3x40x 7, 8p, 10, 11p, 12, 13, 14, 15, 16, ... {10..16} + 7n 3x45x 8, 10, 11p, 12, 13?, 14, 15, 16, 17, ... {10..17} + 8n 3x50x 8, 10, 11p, 12, 13, 14, 15, 16, 17, ... {10..17} + 8n 3x55x 8, 10, 11?, 12, 13, 14, 15, 16, 17, ... {10..17} + 8n 3x60x 7, 8, 10, 11, 12, 13, 14, 15, 16, ... {10..16} + 7n 4x10x ? 4x15x ? 4x20x .. 14p 4x25x .. 14p 4x30x .. 14(p?) 5x5x 12p, ... 12n Impossible: NxZ, NxZx2, 3xNx{3,4,5,6,9} 4xNx{4,5,6,7,8,9} 5xNx{6,7} 3x7xn with n not 0 mod 20, 5x5xn with n not 0 mod 12, 3x10x{7,8,11,12,13,15,16,17,18,19,21,22,23,25,29} 3x15x{7,10,11,13,15} 3x20x{8,11} 3x25x{3..10,11} 4x10x{3..15} 5x8x{8..12} 6x6xN Annotations: Each solutuion of 3*7*N consists of 3*7*20 boxes. 3*7*5n complete: 3*7*20 is unique 3*8*5n complete: 3*8*15 is unique 3*12*5n complete 5*5*n complete: 5*5*12 is unique 3*10*10 and 3*10*14 are unique 3*14*15 has unique central symmetric solution The 5x5xN solutions: 0 0 0 0 0 Start (- is -1 and + is +1) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | + 56t (add 56 T-pentominoes) V - - - 0 - all other 5x5 sections can't be continued for ever. 0 - - - - - - 0 - - I have ommited the reflection. - - - - 0 - 0 - - - |________________ | + 58t | + 4t V V 0 - 0 0 - 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 - 0 + 0 - 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 - 0 0 - 0 0 0 0 0 0 | + 62t V - - - 0 - 0 - - - - - - 0 - - - - - - 0 - 0 - - - So there are fault free solutions for the 5x5x12u box with u odd. the 5x5x12 solution: 3 2 3 3 3 6 8 6 4 6 one half of the box 3 2 2 2 2 6 6 6 6 6 3 2 0 2 3 -> 6 6 6 6 6 2 2 2 2 3 6 6 6 6 6 3 3 3 2 3 6 4 6 8 6 The 3x10x10 box (unique): First found by Fritz G"obel [1] 62 55 55 55 37 44 27 19 12 10 63 65 55 44 44 44 27 27 27 20 64 65 55 45 38 44 27 20 20 20 65 65 65 50 38 38 38 21 22 20 66 66 66 50 38 33 33 33 22 11 67 66 50 50 50 46 33 22 22 22 68 66 61 46 46 46 33 23 23 23 68 68 68 56 39 46 28 29 23 16 68 56 56 56 39 39 39 29 23 17 69 57 51 56 39 43 29 29 29 18 62 62 62 45 37 37 37 19 12 10 63 63 63 45 36 36 36 19 12 10 64 64 64 45 48 36 31 19 12 10 67 54 48 48 48 36 31 21 21 21 67 54 54 54 48 31 31 31 26 11 67 54 49 49 49 32 26 26 26 11 61 61 61 49 42 32 32 32 26 11 69 57 51 49 42 32 28 16 16 16 69 57 51 42 42 42 28 17 17 17 69 57 51 43 43 43 28 18 18 18 62 52 52 52 37 40 24 19 12 10 63 58 52 40 40 40 24 24 24 13 64 58 52 45 34 40 24 13 13 13 58 58 58 47 34 34 34 21 14 13 59 59 59 47 34 30 30 30 14 11 67 59 47 47 47 41 30 14 14 14 60 59 61 41 41 41 30 15 15 15 60 60 60 53 35 41 28 25 15 16 60 53 53 53 35 35 35 25 15 17 69 57 51 53 35 43 25 25 25 18 References: [1] Fritz G"obel; Packing with congruent shapes, CFF 22, 1989, 13-20. (Gives the 3x10x10 packing) [2] Helmut Postl many new prime packings. letter from 9. July 1998 [3] Ross Honsberger; Mathematical Gems II, MAA 1976, The Dolciani Mathematical Expositions 2. (german: Mathematische Juwelen, Vieweg, 1982) Chapter 4: Packing Problems [4] David. A. Klarner; Packing a rectangle with congruent n-ominoes, Journal of Combinatorial Theory 7 (1969) 107-115 [5] David A. Klarner, Frits G"obel; Packing boxes with congruent figures, Indag. Math. 31 (1969) 465-472 [6] Fritz G"obel, Mike Beeler; The T-pentacube packing problem (solutions to problem 1990), Journal of Recreational Mathematics 26 (1994) 66-67 [7] Johan van de Konijnenberg; CFF (2009)