Packing the 5-cube (and 5*5*n boxes) with identical Pentacubes: ------------------------------------- Torsten Sillke, 14.02.96 update w 5.5.21 16.03.96 update w 5.5.23 17.03.96 update w 5.5.n n>=22 18.03.96 update f 5.5.9 no 22.03.96 update f 5.5.n n>=13 22.03.96 update f 5.5.11 26.03.96 update Q 5.5.9 08.06.98 (H. Postl) update 81 5.5.6n 09.07.98 (H. Postl) update glider 09.07.98 (H. Postl) The non-planar pentacubes are numbered with the K"unzell numbers. Notation: p -- indicates a prime-box ? -- open case yet a*b*N -- big N -- indicates the a*b-Strip (one side open) a*b*Z -- big Z -- indicates the a*b-Strip (two sides open) a*b: A, B, C, -- the boxes a*b*A, a*b*B, a*b*C, The 5-cube can be made from identical pentacubes I, L, P (trivial) and Y, N, 41, 71 (non trivial). pentomino I: 5*5: 1, ... n pentomino L: 5*5: 2, 3p, ... {2, 3} + 2n pentomino Y: 5*5: 4p, 5p, 6p, 7p, ... {4..7} + 4n pentomino P: 5*5: 2, 3p, ... {2, 3} + 2n pentomino N: 5*5: 2p, 4, 5p, ... {4, 5} + 2n pentomino T: 5*5: 12p, ... 12*n pentomino V: 5*5: 6p, 9p, 10p, 11p, 12p, 13p, 14p, ... {9..14} + 6n pentomino U: 5*5: 4p, 6, 8, 9, 10, 11p, ... {8..11} + 4n pentomino W: 5*5: 14p, 16p, 18p, 19p, 20p, 21p, 22p, 23p, 24p, 25p, 26p, 27p, 28, 29p, 30, 31p, ... {18..31} + 14n pentomino F: 5*5: 8p, 10p, 11p, 12p, 13p, 14p, 15p, 16, 17p ... {10..17} + 8n pentomino Z: no 5*5*N but boxable pentomino X: not boxable, no N^k pentacube 41: 5*5: 3p, 4p, 5p, ... {3..5} + 3n pentacube 31: 5*5: 4p, 6p, 8, 9p, 10, 11p, ... {8..11} + 4n pentacube 61: 5*5: 2, 4, 6, 8, 9p, ... {8..9} + 2n pentacube 71: 5*5: 4p, 5p, 6, 7p, ... {4..7} + 4n pentacube 81: 5*5: 6p, ... 6n pentacube 82: 5*5: 2p, 4, 6, 8, 10, 11p, ... {10..11} + 2n pentacube 21: no 5*5*N but boxable pentacube 33: no 5*5*N but boxable pentacube 35: not boxable, no N*N*n pentacube 37: no 5*5*N but boxable pentacube 51: no 5*5*N but boxable pseudo-pentomino Q: 5*5: 6p, 9p, ... {6, 9} + 6n pseudo-pentomino glider: 5*5: 4p, 6p, 7p, 8, 9p, ... {6..9} + 4n pseudo-pentomino O: no 5*5*N but boxable pseudo-pentomino large-U: no 5*5*N but boxable ------------------------------------------------------------- Some impossible 5-cubes: -- no 5-cube is possible for pentacubes witch are subsets of the octacubes: 2 2 or 2 2 2 2 2 2. Proof: color the {1,3,5}^3 subset of the {1,2,3,4,5}^3. -- no 5-cube is possible for pentacubes W, Z, Q. Proof: filling a corner forces filling a face-center. -- no 5-cube is possible for polycubes, which have a 4:1 checkerboard coloration. e.g. 1 1 1 1 1 1 1 1 2 1 . 1 1 1 1 Proof: [ 4 1 ] [x] [n+1] [ ] * [ ] = [ ] [ 1 4 ] [y] [ n ] has no integral solution, as x + y = n (modulo 3) and x + y = n+1 (modulo 3). -- no m*n*k box possible for the Q and wide-X pseudo-pentomino if 3 is no divisor of m*n*k. (Proof of H. Postl) Q: 1 wide-X: 1 . 1 large-U: 1 . 1 1 1 . 1 . 1 . 1 1 1 1 . 1 . 1 . 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. -- no 5-cube is possible for polycubes, which have a 4:1 coloration for alternating slices. (Helmut Postl) 81: 1 21: 2 wide-X: 1 . 1 1 1 . 1 . 2 1 1 1 1 . 1 Proof: color the cube (x,y,z) with x (modulo 2). modulo 3 each piece fits each class with 1. Therefore both class must be equal modulo 3. That means, if x is odd then 3 | y*z. ------------------------------------------------------------- Polycubes: Torsten, 1992 ========== A polycube consists of several equal-sized cubes (unit-cubes) joint together such that each cube shares at least one face with another cube. While there is only one monocube (a single unit-cube) and one dicube (two cubes), there are two tricubes (formed by three unit-cubes), eight tetracubes (four cubes), 29 pentacubes (five cubes), 166 hexacubes (six cubes), etc. Notations for polycubes: alpha: height: binary:(Schroeppel) x 1 1 = 1 y 2' 2 = 10 z 2 3 = 11 4 = 100 3" 5 = 101 3' 6 = 110 3 7 = 111 the i-th bit in the binary notation indicates, if there is a cube in the i-th layer. The pentacubes: (the number are the K"unzell Notation) x x x x x x x x x x x x x x x x x x x x 10 11 12 13 x x z z x x x x x x x x x 20 21 22 x x z z x x y y x x x x x x z x x z z x x z z x x x x x x x x z 30 31 32 33 34 35 36 37 x x x x x x x x x x z z x 40 41 42 x x x x x z x x x 50 51 x x x x x x x z x 60 61 x x x x x x x x z z x x x 70 71 72 x x x z x x x x x x x x z 80 81 82 x x x x x 90 The used notation has the advantage over the ascanding numbering, to group the pentacubes. It is derived from their form. The pieces 10, 20, 30, 40, 60, and 70 are easy to recognize, as the form of the piece has similarities with the first digit. The no. 50 reminds at the 5 points on the dice. some pseudo-pentominoes: x x Q x x x x O x x x x x glider (game of life) x x x x x x large U x x x ------------------------------------------------------------- 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 -------------------------------------------------------------------- cube5 of piece 71: (4 solutions) X X 20 20 20 | 33 30 22 17 14 | 33 30 21 21 21 X X 21 20 12 | 33 33 22 17 14 | 33 33 21 20 22 34 31 22 14 11 | 33 25 22 22 15 | 33 25 22 22 22 34 31 22 14 14 | 34 25 24 18 16 | 34 25 24 16 15 34 34 32 14 13 | 34 25 25 18 16 | 34 25 25 16 15 X X 21 21 20 | 33 30 23 14 14 | 33 30 23 21 13 X X 21 12 12 | 32 30 22 17 17 | 32 30 20 20 20 31 31 21 14 11 | 32 32 0 15 15 | 32 32 0 22 20 34 32 22 22 11 | 32 25 24 16 16 | 32 25 24 15 15 32 32 32 13 13 | 34 34 24 18 18 | 34 34 24 16 16 X X 19 15 12 | 30 30 23 19 14 | 30 30 23 17 13 X X 19 19 12 | 32 0 23 17 15 | 32 0 23 13 13 29 31 19 11 11 | 0 0 0 13 15 | 0 0 0 14 13 29 29 22 24 13 | 31 24 24 13 16 | 31 24 24 14 15 29 24 24 24 13 | 34 29 21 18 12 | 34 29 19 16 12 X X 19 15 10 | 28 23 23 19 11 | 28 23 23 17 11 X X 23 15 17 | 28 28 26 19 20 | 28 28 26 17 18 29 26 17 17 17 | 28 27 20 20 20 | 28 27 18 18 18 26 26 18 18 18 | 31 31 21 13 13 | 31 31 19 14 14 27 26 24 18 16 | 29 29 21 12 12 | 29 29 19 12 12 X X 15 15 10 | 28 26 19 19 11 | 28 26 17 17 11 X X 23 10 10 | 26 26 26 11 11 | 26 26 26 11 11 27 23 23 17 10 | 27 27 27 20 11 | 27 27 27 18 11 27 27 23 16 18 | 31 29 27 13 12 | 31 29 27 14 12 27 26 16 16 16 | 31 29 21 21 12 | 31 29 19 19 12 -------------------------------------------------------------------- cube5 of piece 41: (260 solutions) 33 33 24 13 13 33 26 26 26 13 34 27 22 14 14 34 27 22 22 14 34 27 27 15 14 31 26 24 13 21 33 26 21 21 21 34 34 10 14 15 10 10 10 22 15 32 32 27 15 15 31 24 24 13 21 33 24 23 16 12 32 30 10 19 12 32 29 20 22 12 32 25 20 20 11 31 28 18 18 18 31 31 23 16 17 30 30 19 19 17 30 29 19 12 12 30 25 19 20 11 28 28 16 16 18 28 23 23 16 18 28 23 17 17 17 29 29 25 11 11 29 25 25 20 11 -------------------------------------------------------------------- cube5 of piece Y (1264 solutions) 0 0 0 0 12 | . . . . . | . . . . . 29 0 X4 17 14 | . . 21 . . | . 26 . 15 . 29 X1 X4 14 14 | . 24 Y 15 . | . . Y . . 29 25 X4 15 14 | . . 17 . . | . 28 . 17 . 29 21 X4 16 14 | . . . . . | . . . . . 0 0 0 0 12 | H1 26 16 15 11 | 25 25 25 25 13 26 0 17 17 H1 | H1 24 21 15 12 | 26 26 25 15 H1 X4 X4 X4 X4 H1 | H1 24 Y 15 12 | 27 Y Y Y Y 29 25 X1 15 H1 | H1 24 17 15 12 | 28 28 17 17 H1 28 21 21 16 H1 | 29 24 17 23 12 | 29 29 29 29 14 0 0 0 0 12 | 26 26 16 14 11 | 24 24 24 24 13 26 0 X4 17 12 | H1 21 21 14 11 | 27 26 19 15 H1 26 X1 X4 15 H1 | H1 Y Y 14 14 | 27 19 19 15 H1 25 25 X4 15 H1 | 29 22 17 14 12 | 27 28 19 17 H1 28 21 X4 16 16 | 29 23 23 23 23 | 27 29 19 14 14 0 0 0 0 12 | H1 26 16 16 11 | 22 24 16 13 13 26 0 24 17 H1 | H1 22 21 13 10 | 22 26 16 15 H1 24 24 24 24 H1 | H1 22 Y 13 10 | 22 22 16 16 H1 28 25 23 15 H1 | H1 22 17 13 10 | 22 28 16 17 H1 28 21 19 16 H1 | 29 22 20 13 10 | 23 23 23 23 14 0 0 0 0 13 | 25 26 16 18 11 | 20 20 20 20 13 26 0 22 13 13 | 25 18 18 18 18 | H2 20 H2 12 H1 22 22 22 22 13 | 25 25 19 13 10 | H2 H2 H2 12 12 23 23 23 23 13 | 25 19 19 19 19 | H2 H2 H2 12 H1 28 19 19 19 19 | 29 20 20 20 20 | H2 23 H2 12 14 -------------------------------------------------------------------- 88 81 75 69 69 69 5x5x6 box with 88 88 75 75 75 69 89 82 82 82 74 63 89 82 76 76 76 63 1 89 89 76 80 63 63 2 1 1 86 81 81 81 72 69 88 81 75 68 68 68 87 82 74 74 74 68 87 87 76 80 74 67 89 80 80 80 62 63 86 78 72 72 72 61 88 78 78 66 72 68 84 79 79 66 66 66 87 79 73 67 67 67 85 79 73 73 62 67 86 86 83 64 61 61 86 78 70 70 70 61 84 79 71 66 70 61 87 77 77 77 62 60 85 77 73 65 62 62 83 83 83 64 64 64 84 78 83 64 70 60 84 84 71 71 71 60 85 77 71 65 60 60 85 85 73 65 65 65 -------------------------------------------------------------------- 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) - C. J. Bouwkamp, the cube-y problem Cubism for fun 25 ( = CFF 25 silver aniversary ) (dec 1990 - jan 1991) part 3, pp. 30-43. the 1264 distinct solutions to the 5 x 5 x 5 cube. - H. Postl Pentacube 35 is not boxable. The 5*5*n boxes of the Q pseudo pentomino. letter from 8. June 1998 - H. Postl many new prime packings. letter from 9. July 1998 -- mailto:Torsten.Sillke@uni-bielefeld.de http://www.mathematik.uni-bielefeld.de/~sillke/