prime boxes of the n-pentomino 3-D complete: jan. 1998 (the 'p' indicates a prime box) -- computed by Helmut Postl -- 1 1 1 1 1 planar figures: N*N (the quadrant) is possible as the 2*Z bent strip is possible. 2*Z (the strip) is possible. u*Z for u odd: 3 and 5 impossible. ??? what is the smallest u possible ??? 3-Boxes (20 prime boxes): 2x5x 4p, 5p, 6p, 7p, ... {4..7}+4n 3x5x 8p, 12p, 13p, 14p, 15p, 16, 17p, 18p, 19p, ... {12..19}+8n 4x5x 2p, 4, 5, ... {4, 5}+2n 5x5x 2p, 4, 5p, ... {4, 5}+2n 6x5x 2p, 4, 5, ... {4, 5}+2n 7x5x 2p, 4, 5, ... {4, 5}+2n 2x10x 4, 5, 6, 7, ... {4..7}+4n 3x10x 4p, 6p, 7p, 8, 9p, ... {6..9}+4n 4x10x 4, 5, 6, 7, ... {4..7}+4n 5x10x 2, 4, 5, ... {4, 5}+2n 2x15x 4, 5, 6, 7, ... {4..7}+4n 3x15x 4p, 5p, 6p, 7p, ... {4..7}+4n Impossible: Nxk no strips on side open of width k Zx{3,5} 2x2xN, 2x3xN, 3x3xN 3x5x{4, 5, 6, 7, 9, 10, 11} Annotations: All four solutions for packing the 5-cube with the N-pentomino. ---------------------------------------- Torsten Sillke, 14.02.96 58 58 58 48 48 | 59 50 50 43 40 | 59 47 43 42 38 | 58 59 56 56 38 59 59 58 58 39 | 59 59 50 50 50 | 59 47 43 42 39 | 58 59 44 44 44 60 59 59 59 39 | 60 59 48 48 48 | 59 59 43 42 39 | 59 59 45 45 45 60 49 49 43 39 | 60 59 51 51 51 | 60 59 48 48 48 | 59 47 47 46 46 60 50 49 49 49 | 60 51 51 41 38 | 60 48 48 46 46 | 60 60 47 47 47 55 48 48 48 42 | 56 53 43 43 40 | 57 57 57 41 38 | 56 56 56 40 38 60 53 53 53 38 | 60 47 47 47 37 | 60 47 57 57 38 | 58 44 44 41 38 60 54 54 54 38 | 60 48 48 47 47 | 60 47 43 42 39 | 58 45 45 41 41 56 56 56 43 39 | 57 49 58 58 38 | 60 47 43 42 39 | 58 46 46 46 41 57 50 56 56 39 | 58 58 58 41 38 | 58 46 46 46 39 | 57 60 60 60 41 55 47 47 42 42 | 56 53 43 40 40 | 54 54 54 41 41 | 53 53 53 40 40 53 53 47 47 47 | 56 46 46 46 37 | 55 55 55 40 38 | 54 54 54 39 38 54 54 0 0 38 | 0 0 0 46 46 | 0 0 0 40 40 | 0 0 0 39 39 0 0 0 43 38 | 57 49 0 0 38 | 56 56 0 0 40 | 55 55 0 0 39 57 50 50 43 38 | 55 55 55 41 41 | 58 56 56 56 40 | 57 55 55 55 39 55 55 46 42 37 | 53 53 43 40 37 | 51 51 54 54 41 | 50 50 53 53 40 52 46 46 41 37 | 56 45 45 45 37 | 52 52 55 55 38 | 51 51 54 54 38 52 46 44 41 37 | 54 54 54 45 45 | 53 53 53 44 44 | 52 52 52 42 42 52 46 44 41 41 | 57 49 54 54 38 | 58 44 44 44 37 | 57 42 42 42 37 57 57 50 43 41 | 57 49 55 55 41 | 58 45 45 45 37 | 57 43 43 43 37 52 55 44 42 40 | 53 52 42 39 37 | 49 51 51 51 41 | 48 50 50 50 40 52 51 44 40 40 | 56 52 42 39 39 | 49 52 52 52 37 | 48 51 51 51 37 51 51 44 40 37 | 52 52 42 42 39 | 49 49 53 53 37 | 48 48 52 52 37 51 45 45 40 37 | 52 44 44 42 39 | 58 49 50 50 37 | 57 48 49 49 37 51 57 45 45 45 | 57 49 44 44 44 | 50 50 50 45 45 | 49 49 49 43 43 ------------------------------------------------------------------- The Zx11 strip: . 96 96 96 89 89 89 83 83 83 76 76 72 72 72 . . . . 97 96 96 87 89 89 84 83 83 76 76 76 72 72 . . 97 97 91 90 87 87 84 84 78 78 78 73 68 68 68 . . 97 95 91 90 90 87 84 78 78 79 79 73 69 69 68 68 . 97 95 91 91 90 87 84 79 79 79 74 73 73 69 69 69 . . 95 95 91 90 85 85 80 80 80 74 74 73 70 70 70 . . 98 95 92 92 92 85 85 85 80 80 74 70 70 67 . . . 98 92 92 93 93 86 86 86 81 81 74 75 75 67 . . 98 98 93 93 93 86 86 81 81 81 75 75 75 67 67 . . 98 99 99 94 94 94 88 88 82 82 82 77 77 67 71 71 99 99 99 94 94 88 88 88 82 82 77 77 77 71 71 71 . References: - D. A. Klarner A Search for N-Pentacube Prime Boxes, JoRM 12:4 (1979-80) 252-257 (one solution for the 5-cube given) - Helmut Postl Letter from 26. Jan. 1998, complete list of N-pentacube prime boxes Address: Sch"uttaustr. 16/13/7, A-1223 Wien, Austria