Pseudo-Pentomino: Large-U Torsten Sillke, FRA 1998-06 initial version x . x x . x x 2-dim: ------ Z: 5p 3-dim: ------ Nx3: 4, 5p, 6, 7? Nx4: 3, 5p, 6, 7? Nx5: 3p, 4, 6, 7, 8, ... {6..8}+3n 3x10: 4p, 6p, 3x15: 6p, 3x20: 4, 6, 3x25: 4p, 6, 4x5 : 6p, 12, 18, ... 6n ? 4x10: 3p, ... 3n 4x15: 6, 7? 4x20: 3, ... 3n 4x25: 3p, ... 3n Impossible: Zx{3, 4} NxN 3x3xN 4x4xN 5x5xN 3x5xk 3x4x{15} 4x5x{9,15,21} ??? 4x5x3u with u odd ??? Thm: 3 | volume. Proof: color the cube (x,y,z) with x+y+z (modulo 3). The large-U piece covers an odd number of cubes of each color. Therefore the parity of the three colors in the box must be the same. Annotation: 1 2 1 1 1 2 2 1 2 1 2 1 2 1 2 2 1 1 2 1 2 1 2 1 2 1 1 2 2 2 cage symmetry: . . . . . 1 1 . . . . . . 2 2 . 1 . . 2 . 2 2 . . . . . . 1 1 . . . . . the 5xZ strip: a a a a b b b b a a a a b b b b a a c c b b d d c c c c d d d d c c c c d d d d the 4x5x6 box: unique solution (symmetric) T1 T1 77 74 77 74 T2 T2 77 75 77 75 T3 T3 79 77 71 71 H1 79 H1 79 72 72 H2 79 H2 79 73 73 T1 T1 T1 74 66 74 T2 T2 T2 75 67 75 T3 T3 T3 71 68 68 H1 H1 H1 72 69 69 H2 H2 H2 73 70 70 T1 T1 T1 66 74 66 T2 T2 T2 67 75 67 T3 T3 T3 68 71 71 H1 H1 H1 69 72 72 H2 H2 H2 70 73 73 T1 T1 76 66 76 66 T2 T2 76 67 76 67 T3 T3 78 76 68 68 H1 78 H1 78 69 69 H2 78 H2 78 70 70 the 3x4x10 box: (C+, C- block can be rotated.) T1 T1 14 14 11 11 2 2 2 2 T1 T1 T1 13 C+ C- 2 2 2 2 T1 T1 T1 13 C+ C- 2 2 2 2 T1 T1 15 15 2 2 2 2 2 2 H1 14 H1 11 C+ C- 2 2 2 2 H1 H1 H1 C+ C- C+ C- 2 2 2 H1 H1 H1 C+ C- C+ C- 2 2 2 H1 15 H1 13 C+ C- 2 2 2 2 T2 T2 14 14 11 11 2 2 2 2 T2 T2 T2 13 C+ C- 2 2 2 2 T2 T2 T2 13 C+ C- 2 2 2 2 T2 T2 15 15 2 2 2 2 2 2 the 3x6x10 box: 43 38 38 30 30 2 2 2 2 2 43 39 36 36 29 29 2 2 2 2 44 39 34 34 28 2 2 2 2 2 44 40 35 31 28 2 2 2 2 2 45 40 35 31 2 2 2 2 2 2 45 42 42 32 32 2 2 2 2 2 41 39 41 38 2 30 2 2 2 2 41 36 41 29 28 2 2 2 2 2 43 41 35 31 34 2 2 2 2 2 45 37 33 37 33 2 2 2 2 2 44 37 33 37 33 2 2 2 2 2 42 40 37 33 2 32 2 2 2 2 43 38 38 30 30 2 2 2 2 2 43 39 36 36 29 29 2 2 2 2 44 39 34 34 28 2 2 2 2 2 44 40 35 31 28 2 2 2 2 2 45 40 35 31 2 2 2 2 2 2 45 42 42 32 32 2 2 2 2 2 the 3x6x15 box: 33 33 24 24 21 13 13 2 2 2 2 2 2 2 2 34 29 29 21 18 21 18 2 2 2 2 2 2 2 2 34 30 30 21 18 21 18 2 2 2 2 2 2 2 2 35 31 31 22 22 18 14 14 2 2 2 2 2 2 2 35 32 32 23 23 14 12 12 2 2 2 2 2 2 2 36 36 25 25 19 19 14 14 2 2 2 2 2 2 2 34 26 33 26 24 16 11 13 2 2 2 2 2 2 2 29 26 27 26 16 11 16 11 2 2 2 2 2 2 2 30 27 26 27 16 11 16 11 2 2 2 2 2 2 2 31 27 28 27 17 22 17 2 2 2 2 2 2 2 2 32 28 23 28 17 12 17 2 2 2 2 2 2 2 2 35 28 36 28 25 17 19 2 2 2 2 2 2 2 2 33 33 24 24 20 13 13 2 2 2 2 2 2 2 2 34 29 29 20 15 20 15 2 2 2 2 2 2 2 2 34 30 30 20 15 20 15 2 2 2 2 2 2 2 2 35 31 31 22 22 15 10 10 2 2 2 2 2 2 2 35 32 32 23 23 10 12 12 2 2 2 2 2 2 2 36 36 25 25 19 19 10 10 2 2 2 2 2 2 2 the 3x4x25 box: 68 68 62 62 61 55 55 50 50 46 46 41 41 36 36 31 31 26 26 21 21 16 16 14 14 69 69 68 61 56 61 53 51 48 48 43 43 38 37 34 32 32 27 24 27 19 17 14 15 15 68 68 69 61 56 61 53 51 49 49 44 44 38 37 34 33 33 27 24 27 19 17 15 14 14 69 69 63 63 57 57 54 54 47 47 45 45 40 40 35 35 30 30 27 23 23 18 18 15 15 66 62 66 59 55 59 53 52 46 50 41 42 39 37 31 36 28 28 24 26 20 21 12 16 12 66 67 66 59 60 59 52 48 52 43 42 39 42 39 32 28 29 29 22 20 22 20 12 13 12 67 66 67 60 59 60 52 49 52 44 42 39 42 39 33 29 28 28 22 20 22 20 13 12 13 67 63 67 60 56 60 57 51 54 45 47 40 38 35 34 30 29 29 23 22 19 17 13 18 13 64 64 62 62 58 55 55 50 50 46 46 41 41 36 36 31 31 26 26 21 21 16 16 10 10 65 65 64 58 56 58 53 51 48 48 43 43 38 37 34 32 32 25 24 25 19 17 10 11 11 64 64 65 58 56 58 53 51 49 49 44 44 38 37 34 33 33 25 24 25 19 17 11 10 10 65 65 63 63 57 57 54 54 47 47 45 45 40 40 35 35 30 30 25 23 23 18 18 11 11 the 3x5xN strip: unique solution . . . 34 34 26 26 21 21 14 14 . . 35 . . . 32 32 25 25 19 19 13 13 . 35 30 . . . 27 27 22 22 15 15 10 10 36 30 . . . 29 29 23 23 17 17 11 11 36 . . . 31 31 24 24 18 18 12 12 . 35 . . . 28 34 28 26 16 21 16 14 . . . . 32 28 25 28 19 16 13 16 . . 36 . . . 33 28 22 27 20 16 10 15 . . . . 33 29 33 23 20 17 20 11 . . . 30 . 33 . 33 31 20 24 20 18 . 12 . . . 34 34 26 26 21 21 14 14 . . 35 . . . 32 32 25 25 19 19 13 13 . 35 30 . . . 27 27 22 22 15 15 10 10 36 30 . . . 29 29 23 23 17 17 11 11 36 . . . 31 31 24 24 18 18 12 12 .