Prime Packings of the Pseudo-Pentomino Q: Torsten Sillke, FRA 3-dim complete: 16.06.98 Correction: rectangles possible[2], 2-dim almost complete: H. Postl Update: (5+6n)xZ, 14xZ stripes 1998-07-21 Update: 30x30 impossible 1998-07-26 2-dim complete, 1998-07-26 Correction: 24x30p: Mike Reid 1999-01 Open: 12xZ stripes 1 1 1 1 1 2-dim: ------ Zx 5p, 8p, 10, 11p, 13, 14p, 15, 16, 17p, ... {13..17}+5n Nx 16, 18, 20, 22, 24, 26, 28, 30, ... {16,18,..,30}+16n 20x 24p, 30p, 36p, 42p, ... {24,30,36,42}+24n 30x 16p, 20p, 24p, 26p, 28p, 32, 34p, 36, 38p, ... {24,28,...,38}+16p 40x 18p, 24, 30, ... {18,24,30}+18n 50x 18p, 24, 30, ... {18,24,30}+18n 60x 16, 20, 24, 26, 28, 30, 32, 34, 36, 38, ... {24,26,28,...,38}+16n 70x 24, 30, 36, 42, ... {24,30,36,42}+24n 80x 18, 24, 30, ... {18,24,30}+18n 90x 16, 18, 20, 22p, 24, 26, 28, 30, ... {16,18,...,30}+16n 120x ... 22p ... 150x ... 22p ... 3-dim: ------ 3x5x 6p, 12, 14p, 15p, 16p, 17p, 18, 19p, ... {14..19}+6n 4x5x 6p, 9p, ... {6, 9}+6n 5x5x 6p, 9p, ... {6, 9}+6n 6x5x 3p, 4p, 5p, ... {3..5}+3n 7x5x 6, 9p, ... {6, 9}+6n 8x5x 6, 9, ... {6, 9}+6n 9x5x 4p, 5p, 6, 7p, ... {4..7}+4n 3x10x 3p, 4p, 6, 7, 8, ... {6..8}+3n 4x10x 3p, ... 3n 5x10x 6, 9, ... {6, 9}+6n 6x10x 3, 4, 5, ... {3..5}+3n 3x15x 3p, 4p, 5p, ... {3..5}+3n 4x15x 3p, 4p, 5, ... {3..5}+3n 5x15x 3p, 4, 5, ... {3..5}+3n Complete list of prime rectangles (# = 15): 16x30, 18x{40,50}, 20x{24,30,36,42}, 22x{90,120,150}, 24x30, 26x30, 28x30, 30x{34,38} Complete list of prime boxes (# = 16): 3x3x{10,15}, 3x4x{10,15}, 3x5x{6,14,15,16,17,19}, 4x4x15, 4x5x{6,9}, 5x5x{6,9}, 5x7x9 Impossible: Zx{3, 4, 6, 7, 9, 12? } Nx{8, 10, 12, 14} 18x{20,30,60,70,110} 22x{30,60} 3x5x{3, 4, 5, 7, 8, 9, 10, 11, 13} Thm[1,2]: impossibility of boxes with Volume != 0 (modulo 3) (Proof of H. Postl, June 1998) Proof: color the cube (x,y,z) with x+y+z (modulo 3). The Q piece covers an odd number of cubes of each color. Therefore the parity of the three colors in the box must be the same. Thm: Every tiling of the halve plane NxZ has no single square at the border. Proof: trivial case checking. . . . 1 1 . . . . 1 1 2 2 . . . . 1 1 . -> fill a -> . . . 1 1 2 2 . -> filling b is impossible. . . 1 . a . . . 1 . 2 b . . ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ Cor[2]: both sides of a rectangle are even. Thm: no 7xZ Proof: the borders are filled with 2x2 squares. Now the third and fifth row are half filled with single squares while the middle row is still empty. Middle row : Non-Middle row space is 1 : 1. But each Q-piece fills 2:3. As the pieces have bounded size this is impossible. Thm: no 9xZ Proof: the borders are filled with 2x2 squares. The density of free squares in the middle rows is (d1,d2,d3,d4,d5) = (1/2, 1, 1, 1, 1/2). Now fold the strip (d1+d5, d2+d3, d3) = (1, 2, 1). Each Q-piece can fill squares of these three types in three different ways. To calculata the proportions we find the following linear system: [d1+d5] [1] [2 0 1] [x1] [d2+d4] = [2] = [2 3 2] * [x2] [ d3 ] [1] [1 2 2] [x3] This system has the unique solutuion (x1, x2, x3) = (3, 2, -1)/5. As x3 = -1/5 is negative no solution can be possible. Annotations: two Q give the following decominoes 2 1 1 1 1 1 1 1 2 2 1 1 2 1 1 2 2 1 1 . 2 2 1 1 2 2 1 2 2 1 1 2 2 1 1 . 2 2 2 2 2 2 the NxN octant: all solutions have the following corner where the 'x' marked places are filled with a 2-square. . . . 5 5 . . . . 5 5 3 x x . 3 3 5 x x . 3 3 1 4 2 . . 1 1 2 2 4 4 . 1 1 2 2 4 4 . the 5xZ strip: a a b b a a b b a a b b a a b b a a b b a a b b a a b b a a b b a a b b a a b b a a b b a a b b a a b b a a b b a a b b the 8xZ strip: a a a a b b b b 1 1 a a a a b b b b 1 1 a a a a b b b b 1 1 a a a a b b b b 1 1 a a c c b b 2 c c 1 a a c c b b 2 c c 1 2 2 c c c 2 2 c c c 2 2 c c c 2 2 c c c 2 2 c c c 2 2 c c c 2 2 c c c 2 2 c c c b b 2 c c 1 a a c c b b 2 c c 1 a a c c b b b b 1 1 a a a a b b b b 1 1 a a a a b b b b 1 1 a a a a b b b b 1 1 a a a a the (5+6n)xZ strip: (period 11) a a a a 2 2 b b b b a a a a 2 2 b b b b . . a a 2 . . . b b b b . . . a a 1 . . b b b 1 1 a a a 1 1 b b b 1 1 a a a 1 1 b b 1 a a a a 2 b b 2 2 a a a 2 2 b b b 2 2 a a a 2 2 b b b . . 2 a a . . . b b b b . . . 1 a a . . b b b b 1 1 a a a a b b b b 1 1 a a a a the 14xZ strip: (period 21) . 60 60 57 57 50 50 43 43 38 38 35 35 26 26 25 25 16 16 12 12 . . 60 60 57 57 50 50 43 43 38 38 35 35 26 26 25 25 16 16 12 12 . . . 57 60 50 52 52 44 44 43 35 38 36 27 25 26 19 19 12 16 11 11 . . 58 58 51 52 52 44 44 45 36 36 29 29 27 27 19 19 20 10 11 11 . . 58 58 52 51 51 45 45 44 36 36 29 29 27 27 20 20 19 11 10 10 . . 62 62 58 51 51 45 45 42 42 33 33 28 29 21 20 20 13 13 10 10 . 61 62 62 53 53 49 49 41 42 42 33 33 34 28 28 21 21 13 13 14 . . 62 61 61 53 53 49 49 42 41 41 34 34 33 28 28 21 21 14 14 13 . 64 64 61 61 54 49 53 46 46 41 41 34 34 31 31 22 22 17 14 14 . . 64 64 65 56 56 54 54 46 46 47 39 39 30 31 31 22 22 23 17 17 . . 65 65 64 56 56 54 54 47 47 46 39 39 31 30 30 23 23 22 17 17 . . 65 65 59 63 55 56 48 47 47 39 37 40 32 30 30 23 23 24 15 18 . . . 63 63 59 59 55 55 48 48 40 40 37 37 32 32 24 24 18 18 15 15 . . 63 63 59 59 55 55 48 48 40 40 37 37 32 32 24 24 18 18 15 15 . the 16x30 rectangle: (several similar solutions) - - - - - - 70 70 63 63 58 58 52 52 44 44 38 38 33 33 26 26 24 24 - - - - - - - - - - - - 70 70 63 63 58 58 52 52 44 44 38 38 33 33 26 26 24 24 - - - - - - - - - - - 78 73 73 70 61 63 54 58 48 52 42 44 35 38 26 33 24 18 18 14 - - - - - - - - 78 78 72 73 73 64 64 61 61 54 54 48 48 42 42 35 35 27 27 18 18 19 14 14 - - - - - - 78 78 73 72 72 64 64 61 61 54 54 48 48 42 42 35 35 27 27 19 19 18 14 14 - - - - - 84 79 82 74 72 72 65 65 64 55 55 49 49 45 45 39 39 27 29 29 19 19 20 12 15 10 - - 84 84 82 82 79 79 74 74 65 65 59 55 55 49 49 45 45 39 39 28 29 29 20 20 15 15 12 12 10 10 84 84 82 82 79 79 74 74 66 66 65 59 59 55 45 49 39 36 36 29 28 28 20 20 15 15 12 12 10 10 85 85 83 83 80 80 75 75 66 66 67 59 59 56 46 50 40 36 36 30 28 28 21 21 16 16 13 13 11 11 85 85 83 83 80 80 75 75 67 67 66 56 56 50 50 46 46 40 40 36 30 30 21 21 16 16 13 13 11 11 - - 85 80 83 75 77 77 67 67 68 56 56 50 50 46 46 40 40 31 30 30 22 22 21 13 16 11 - - - - - 81 81 76 77 77 68 68 60 60 53 53 47 47 41 41 34 34 31 31 22 22 23 17 17 - - - - - - 81 81 77 76 76 68 68 60 60 53 53 47 47 41 41 34 34 31 31 23 23 22 17 17 - - - - - - - - 81 76 76 71 62 69 57 60 51 53 43 47 37 41 32 34 25 23 23 17 - - - - - - - - - - - 71 71 69 69 62 62 57 57 51 51 43 43 37 37 32 32 25 25 - - - - - - - - - - - - 71 71 69 69 62 62 57 57 51 51 43 43 37 37 32 32 25 25 - - - - - - the 20x24 rectangle: (several similar solutions) - - - - - - 26 26 22 22 14 14 2 2 2 2 2 2 - - - - - - - - - - - - 26 26 22 22 14 14 2 2 2 2 2 2 - - - - - - - - - - - 30 30 22 26 14 10 10 2 2 2 2 2 2 2 - - - - - - - - 40 40 30 30 31 23 23 10 10 2 2 2 2 2 2 2 2 2 - - - - - - 40 40 31 31 30 23 23 18 2 10 2 2 2 2 2 2 2 2 - - - - - 40 36 41 31 31 23 18 18 15 15 2 2 2 2 2 2 2 2 2 2 - - 44 44 41 41 36 36 32 32 18 18 15 15 2 2 2 2 2 2 2 2 2 2 2 2 44 44 41 41 36 36 32 32 27 15 11 11 2 2 2 2 2 2 2 2 2 2 2 2 45 45 44 37 37 32 27 27 19 19 11 11 2 2 2 2 2 2 2 2 2 2 2 2 45 45 46 37 37 38 27 27 19 19 20 2 11 2 2 2 2 2 2 2 2 2 2 2 46 46 45 38 38 37 28 28 20 20 19 2 12 2 2 2 2 2 2 2 2 2 2 2 46 46 47 38 38 33 28 28 20 20 12 12 2 2 2 2 2 2 2 2 2 2 2 2 47 47 42 42 39 39 33 33 28 16 12 12 2 2 2 2 2 2 2 2 2 2 2 2 47 47 42 42 39 39 33 33 21 21 16 16 2 2 2 2 2 2 2 2 2 2 2 2 - - 43 39 42 34 34 24 21 21 16 16 2 2 2 2 2 2 2 2 2 2 - - - - - 43 43 34 34 35 24 24 21 2 13 2 2 2 2 2 2 2 2 - - - - - - 43 43 35 35 34 24 24 13 13 2 2 2 2 2 2 2 2 2 - - - - - - - - 35 35 25 29 17 13 13 2 2 2 2 2 2 2 - - - - - - - - - - - 29 29 25 25 17 17 2 2 2 2 2 2 - - - - - - - - - - - - 29 29 25 25 17 17 2 2 2 2 2 2 - - - - - - the 30x24 rectangle: (several similar solutions) + + + + + + 48 48 32 32 + + + + + + 2 2 2 2 2 2 2 2 2 2 2 2 2 2 + + + + + + 48 48 32 32 + + + + + + 2 2 2 2 2 2 2 2 2 2 2 2 2 2 + + + + + 48 50 50 33 33 32 + + + + + 2 2 2 2 2 2 2 2 2 2 2 2 2 2 + + + 56 56 49 50 50 33 33 34 17 17 + + + 2 2 2 2 2 2 2 2 2 2 2 2 2 2 + + + 56 56 50 49 49 34 34 33 17 17 + + + 2 2 2 2 2 2 2 2 2 2 2 2 2 2 + + 73 57 65 56 49 49 34 34 35 18 28 17 + + 2 2 2 2 2 2 2 2 2 2 2 2 2 2 73 73 65 65 57 57 42 42 35 35 28 28 18 18 13 13 2 2 2 2 2 2 2 2 2 2 2 2 2 2 73 73 65 65 57 57 42 42 35 35 28 28 18 18 13 13 2 2 2 2 2 2 2 2 2 2 2 2 2 2 74 74 66 66 58 58 51 51 42 37 37 19 19 13 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 74 74 66 66 58 58 51 51 36 37 37 19 19 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 75 75 74 58 66 51 43 52 37 36 36 20 20 19 14 14 2 2 2 2 2 2 2 2 2 2 2 2 2 2 75 75 76 59 59 52 52 43 43 36 36 20 20 21 14 14 2 2 2 2 2 2 2 2 2 2 2 2 2 2 76 76 75 59 59 52 52 43 43 29 29 21 21 14 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 76 76 77 60 67 59 44 44 38 29 29 21 21 15 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 77 77 67 67 60 60 44 44 45 38 38 29 23 23 15 15 2 2 2 2 2 2 2 2 2 2 2 2 2 2 77 77 67 67 60 60 45 45 44 38 38 22 23 23 15 15 2 2 2 2 2 2 2 2 2 2 2 2 2 2 78 78 68 68 61 61 45 45 46 30 39 23 22 22 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 78 78 68 68 61 61 46 46 39 39 30 30 22 22 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 + + 78 61 68 62 46 46 39 39 30 30 + + 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 + + + 62 62 53 53 47 47 40 40 + + + 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 + + + 62 62 53 53 47 47 40 40 + + + 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 + + + + + 54 47 53 40 + + + + + 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 + + + + + + 54 54 + + + + + + 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 + + + + + + 54 54 + + + + + + 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 the 3x5x6 box: unique 98 98 92 92 88 88 99 94 94 90 90 84 99 94 94 84 84 82 94 91 91 84 84 82 93 93 89 89 83 83 98 98 92 92 88 88 99 95 95 90 90 85 99 95 95 85 85 82 95 91 91 85 85 82 93 93 89 89 83 83 99 92 98 88 86 86 97 97 90 87 86 86 97 97 96 86 87 87 96 96 97 91 87 87 96 96 93 83 89 82 the 3x3x10 box: 97 94 96 95 90 88 88 85 85 82 98 95 95 92 92 90 90 85 85 82 99 95 95 93 93 90 90 87 87 85 96 96 94 94 89 88 88 84 84 82 97 98 98 92 92 89 89 84 84 82 97 99 99 93 93 89 89 87 87 84 96 96 94 94 88 91 86 86 83 83 97 98 98 91 91 92 86 86 83 83 97 99 99 91 91 93 87 83 86 82 the 3x3x15 box: (odd box) 97 94 96 95 90 93 87 87 83 83 79 79 78 78 74 98 95 95 93 93 90 90 85 83 83 80 80 78 78 75 99 95 95 93 93 90 90 85 84 84 83 78 76 77 73 96 96 94 94 89 92 87 87 82 82 79 79 74 74 73 97 98 98 92 92 89 89 85 82 82 80 80 75 75 73 97 99 99 92 92 89 89 85 84 84 82 77 77 76 76 96 96 94 94 88 91 86 85 87 79 81 81 74 74 73 97 98 98 91 91 88 88 86 86 80 81 81 75 75 73 97 99 99 91 91 88 88 86 86 81 84 77 77 76 76 the 3x5x14 box: 98 99 99 86 83 83 78 78 72 72 69 69 64 64 98 94 93 93 86 86 76 76 73 73 66 67 67 60 97 94 92 92 86 86 76 76 73 73 66 60 60 58 97 91 91 89 89 82 82 73 76 68 68 60 60 58 95 95 88 88 84 84 79 79 70 70 65 65 59 59 98 99 99 87 83 83 78 78 72 72 69 69 64 64 98 94 93 93 87 87 77 77 74 74 66 67 67 61 97 94 92 92 87 87 77 77 74 74 66 61 61 58 97 91 91 89 89 82 82 74 77 68 68 61 61 58 95 95 88 88 84 84 79 79 70 70 65 65 59 59 99 94 90 90 85 85 83 75 78 69 72 64 62 62 97 93 90 90 85 85 80 80 75 75 67 63 62 62 98 92 96 85 90 81 80 80 75 75 71 62 63 63 96 96 89 91 82 80 81 81 71 71 66 68 63 63 96 96 95 84 88 79 81 81 71 71 70 59 65 58 the 3x5x15 box: 98 98 91 91 89 81 84 75 73 73 69 69 60 65 58 98 98 91 91 86 86 78 78 73 73 69 69 58 58 59 99 91 98 87 86 86 78 78 77 69 73 61 58 58 55 96 92 88 86 87 87 77 77 78 74 66 68 61 61 56 97 90 95 85 87 87 77 77 76 67 70 62 61 61 57 99 93 89 89 84 84 81 81 75 75 71 65 65 60 60 99 92 93 93 82 82 79 79 71 71 66 63 59 59 56 97 92 93 93 82 82 79 79 71 71 66 62 63 63 56 97 96 96 88 88 79 82 74 74 68 68 62 63 63 55 95 95 90 90 85 85 76 76 70 70 67 67 57 57 55 99 94 89 89 84 84 81 81 75 75 72 65 65 60 60 99 92 94 94 83 83 80 80 72 72 66 64 59 59 56 97 92 94 94 83 83 80 80 72 72 66 62 64 64 56 97 96 96 88 88 80 83 74 74 68 68 62 64 64 55 95 95 90 90 85 85 76 76 70 70 67 67 57 57 55 the 4x4x15 box: 97 97 91 91 83 80 98 98 93 93 88 88 98 98 93 93 88 88 99 93 98 88 84 84 97 97 91 91 82 85 94 96 96 86 83 80 95 96 96 86 83 80 96 99 99 87 84 84 95 91 97 85 85 82 82 x x 95 92 92 86 83 80 x x x 94 92 92 86 83 80 x x x 94 99 99 92 87 87 84 x x 95 89 89 85 85 82 82 y y 95 89 89 90 81 81 y y y 94 90 90 89 81 81 y y y 94 90 90 86 87 87 81 y y the impossible square 30x30: To speed up the computation fill up all corners. The places at the border can be filled in two ways. Let's compute the simpler branch first. The other branch get reduced by the symmetry of the square. As non of the following three cases is solvable the square 30x30 is impossible. Case 1: (2.5 h computation time) - - - - - - r r * * * * * * * * * * * * * * * * - - - - - - - - - - - - r r * * * * * * * * * * * * * * * * - - - - - - - - - - - r * * * * * * * * * * * * * * * * * * * - - - - - - - - * * * * * * * * * * * * * * * * * * * * * * * * - - - - - - * * * * * * * * * * * * * * * * * * * * * * * * - - - - - * * * * * * * * * * * * * * * * * * * * * * * * * * - - * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * - - * * * * * * * * * * * * * * * * * * * * * * * * * * - - - - - * * * * * * * * * * * * * * * * * * * * * * * * - - - - - - * * * * * * * * * * * * * * * * * * * * * * * * - - - - - - - - * * * * * * * * * * * * * * * * * * * * - - - - - - - - - - - * * * * * * * * * * * * * * * * * * - - - - - - - - - - - - * * * * * * * * * * * * * * * * * * - - - - - - Case 2: (70 h computation time) - - - - - - r r s s * * * * * * * * * * * * r r - - - - - - - - - - - - r r s s * * * * * * * * * * * * r r - - - - - - - - - - - * * s r * * * * * * * * * * * * r * * * - - - - - - - - * * * * * * * * * * * * * * * * * * * * * * * * - - - - - - * * * * * * * * * * * * * * * * * * * * * * * * - - - - - * * * * * * * * * * * * * * * * * * * * * * * * * * - - r r * * * * * * * * * * * * * * * * * * * * * * * * * * r r r r * * * * * * * * * * * * * * * * * * * * * * * * * * r r * * r * * * * * * * * * * * * * * * * * * * * * * * * r * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * r * * * * * * * * * * * * * * * * * * * * * * * * r * * r r * * * * * * * * * * * * * * * * * * * * * * * * * * r r r r * * * * * * * * * * * * * * * * * * * * * * * * * * r r - - * * * * * * * * * * * * * * * * * * * * * * * * * * - - - - - * * * * * * * * * * * * * * * * * * * * * * * * - - - - - - * * * * * * * * * * * * * * * * * * * * * * * * - - - - - - - - * * * r * * * * * * * * * * * * r * * * - - - - - - - - - - - r r * * * * * * * * * * * * * * r r - - - - - - - - - - - - r r * * * * * * * * * * * * * * r r - - - - - - Case 3: (45 min. computation time) - - - - - - r r r r * * * * * * * * * * r r r r - - - - - - - - - - - - r r r r * * * * * * * * * * r r r r - - - - - - - - - - - * * * r * r * * * * * * * * r * r * * * - - - - - - - - * * * * * * * * * * * * * * * * * * * * * * * * - - - - - - * * * * * * * * * * * * * * * * * * * * * * * * - - - - - * * * * * * * * * * * * * * * * * * * * * * * * * * - - r r * * * * * * * * * * * * * * * * * * * * * * * * * * r r r r * * * * * * * * * * * * * * * * * * * * * * * * * * r r r r r * * * * * * * * * * * * * * * * * * * * * * * * r r r r r * * * * * * * * * * * * * * * * * * * * * * * * * * r r * * r * * * * * * * * * * * * * * * * * * * * * * * * r * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * r * * * * * * * * * * * * * * * * * * * * * * * * r * * r r * * * * * * * * * * * * * * * * * * * * * * * * * * r r r r r * * * * * * * * * * * * * * * * * * * * * * * * r r r r r * * * * * * * * * * * * * * * * * * * * * * * * * * r r r r * * * * * * * * * * * * * * * * * * * * * * * * * * r r - - * * * * * * * * * * * * * * * * * * * * * * * * * * - - - - - * * * * * * * * * * * * * * * * * * * * * * * * - - - - - - * * * * * * * * * * * * * * * * * * * * * * * * - - - - - - - - * * * r * r * * * * * * * * r * r * * * - - - - - - - - - - - r r r r * * * * * * * * * * r r r r - - - - - - - - - - - - r r r r * * * * * * * * * * r r r r - - - - - - References: [1] H. Postl Pentacube 35 is not boxable. The 5*5*n boxes of the Q pseudo pentomino. letter from 8. June 1998 [2] H. Postl many new prime packings. letter from 9. July 1998