prime boxes of the u-pentomino (:.:) Torsten Sillke, complete: 1993-01-08 (the 'p' indicates a prime box) 2-dim: ZxZ 3-dim: 2x5x 3p, 6, 9, ... 3n 3x5x 2p, 4, 6, 7p, ... {6..7}+2n 4x5x 3, 4p, 5p, ... {3..5}+3n 5x5x 4p, 6, 8, 9, 10, 11p, ... {8..11}+4n 6x5x 2, 3, ... {2..3}+2n 7x5x 3p, 4, 6, ... {3..6}+3n 2x10x 3, 6, 9, 10p, 12, 13, 14p, ... {12..14}+3n 3x10x 2, 3p, ... {2..3}+2n 4x10x 3, 4, 5, ... {3..5}+3n 5x10x 3, 4, 5, ... {3..5}+3n 2x15x 3, 5, 6, 8, 9, 10, ... {8..10}+2n 3x15x 2, 4, 5, ... {4..5}+2n 4x15x 3, 4, 5, ... {3..5}+3n 5x15x 2, 3, ... {2..3}+2n 2x20x 3, 6, 9, 10, 11p, {9..11}+3n 2x25x 3, 6. 9, 10, 11p, {9..11}+3n Complete list of prime boxes (# = 10): 2x3x5 2x10x{10,14} 2x11x{20,25} 3x3x10 3x5x7 4x4x5 4x5x5 5x5x11 4-dim: 2x2x5x 3, 4p, 6, 7, 8, ... {6..8}+3n 2x3x5x 1p, ... n 3x3x5x 2, 3p, ... {2..3}+2n 2x5x5x 3, 4, 5p, ... {3..5}+3n 3x5x5x 2, 3, ... {2..3}+2n Complete list of prime hyperboxes (# = 3): 2x2x4x5, 2x5x5x5, 3x3x3x5 Impossible: ZxN 2x5xk (k = 1,2 mod 3) 2x8x5k (k = 1,2 mod 3) 3x3x5u (u odd) 5x5x{3, 5, 7} 2x4xN 2x7xN 2x10x11 2x2...2xN 2x2...2x5x5 Annotations: There is one solutions for the 2x4x5 box with U and X pentominos. a b c b c a x c c c A 2x4x5 box with X and U. a b b b d x x x y d a e e e d a x y y y f e f e d f f f y d This gives solutions for the 4x4x5 and 2x2x4x5 boxes with the u only. There is a (u, x) solutions for the 5x5x5 cube (2 (2x4x5) & 3 (3x5)). a a a x x two 2x4 and three 1x3 form a 5x5. y y b x x y y b x x y y b x x y y c c c This gives a solution for the 2x5x5x5 box with the u only. The X-symmetry: 2 2 2 2 2 the 2x3x5 box: u u x u u u x x x u u u x u u the 2x5xN strip: This strip dissects into 2x3x5 boxes. the 2x8x15n boxes: All solutions of the 2x8x15n boxes consist of 2x3x5 boxes. The 2x8x15n boxes dissect into 2x8x15 boxes which are unique. a a a a a b b b b b a a a a a a a a a a b b b b b a a a a a a a a a a b b b b b a a a a a b b b c c c a a a c c c b b b b b b c c c a a a c c c b b b b b b c c c a a a c c c b b b b b b c c c a a a c c c b b b b b b c c c a a a c c c b b b the 2x10x(10+4n) box: 45 45 45 32 28 32 A A A A A 20 20 20 10 46 40 40 32 32 32 A A A A A 21 21 21 10 46 40 36 36 28 A A A A A 25 21 25 21 10 46 40 40 36 A A A A A 29 26 29 26 14 14 48 48 36 36 A A A A A 29 29 29 15 14 15 47 48 47 33 33 33 B B B B B 22 22 14 14 48 48 37 33 37 33 B B B B B 22 16 16 11 49 41 38 41 38 B B B B B 27 22 22 16 11 49 41 41 41 B B B B B 30 30 30 16 16 11 49 42 42 42 B B B B B 30 27 30 17 17 17 45 39 45 39 28 A A A A A 23 20 23 20 10 46 39 39 39 28 A A A A A 23 23 23 12 12 43 43 34 34 28 A A A A A 25 25 25 12 10 46 43 34 31 31 A A A A A 26 26 26 12 12 43 43 34 34 31 A A A A A 24 24 15 15 15 47 47 47 31 31 B B B B B 24 18 18 13 13 44 44 37 37 37 B B B B B 24 24 18 13 11 49 44 38 38 38 B B B B B 27 18 18 13 13 44 44 35 35 35 B B B B B 27 19 19 19 11 49 42 35 42 35 B B B B B 27 19 17 19 17 with the plus 4 extension A equal B 22 11 23 11 22 11 11 11 20 22 21 23 10 10 10 21 10 20 10 21 13 22 13 23 13 13 13 23 20 22 21 23 20 12 12 12 20 12 21 12 the 2x11x20 box: 47 44 47 44 31 25 22 25 12 12 12 2 2 2 2 2 2 2 2 2 48 44 44 44 31 25 25 25 13 13 13 2 2 2 2 2 2 2 2 2 49 49 40 37 31 26 22 18 13 18 13 2 2 2 2 2 2 2 2 2 48 49 40 37 32 32 32 18 18 18 2 2 2 2 2 2 2 2 2 2 49 49 40 37 32 26 32 19 19 2 2 2 2 2 2 2 2 2 2 2 50 50 50 38 38 27 23 23 19 2 2 2 2 2 2 2 2 2 2 2 51 51 41 41 38 27 23 19 19 10 10 2 2 2 2 2 2 2 2 2 52 51 41 38 38 27 23 23 14 14 10 2 2 2 2 2 2 2 2 2 51 51 41 41 33 28 28 28 14 10 10 2 2 2 2 2 2 2 2 2 52 45 45 45 33 29 29 29 14 14 2 2 2 2 2 2 2 2 2 2 53 45 53 45 33 29 24 29 24 2 2 2 2 2 2 2 2 2 2 2 47 47 47 34 31 34 22 15 12 15 12 2 2 2 2 2 2 2 2 2 48 42 42 34 34 34 22 15 15 15 2 2 2 2 2 2 2 2 2 2 48 42 40 37 31 26 22 16 16 2 2 2 2 2 2 2 2 2 2 2 48 42 42 35 35 26 20 20 16 2 2 2 2 2 2 2 2 2 2 2 46 46 40 37 35 26 20 16 16 2 2 2 2 2 2 2 2 2 2 2 50 46 50 35 35 27 20 20 11 11 11 2 2 2 2 2 2 2 2 2 46 46 39 39 30 30 30 17 11 17 11 2 2 2 2 2 2 2 2 2 52 43 43 39 30 27 30 17 17 17 2 2 2 2 2 2 2 2 2 2 52 43 39 39 33 28 21 28 21 2 2 2 2 2 2 2 2 2 2 2 52 43 43 36 36 36 21 21 21 2 2 2 2 2 2 2 2 2 2 2 53 53 53 36 33 36 24 24 24 2 2 2 2 2 2 2 2 2 2 2 65 65 55 51 51 51 38 32 38 22 22 22 X C C A A A A A 65 59 59 59 45 45 38 38 38 24 24 X X X C A A A A A 65 65 55 52 45 52 39 32 39 23 24 23 X C C A A A A A 66 60 66 60 45 45 39 39 39 24 24 A A A B B B C C C 67 60 60 60 46 46 46 33 29 29 29 A X A B B B C C C 68 68 56 53 53 42 40 40 40 25 25 X X X B B B C C C 67 68 56 53 47 47 47 33 30 25 30 A X A B B B C C C 68 68 56 53 53 42 41 41 41 25 25 A A A B B B C C C 69 61 57 61 57 43 41 34 41 26 26 26 X C C A A A A A 69 61 61 61 48 48 48 34 31 31 31 X X X C A A A A A 69 62 62 62 48 43 48 34 31 27 31 27 X C C A A A A A 63 63 55 51 44 51 44 32 28 22 28 22 X C C A A A A A 63 59 55 59 44 44 44 32 28 28 28 X X X C A A A A A 63 63 55 52 52 52 35 32 35 23 23 23 X C C A A A A A 66 66 66 49 49 49 35 35 35 20 20 A A A B B B C C C 67 58 58 49 46 49 46 33 29 20 29 A X A B B B C C C 67 58 56 50 50 42 40 33 40 20 20 X X X B B B C C C 67 58 58 50 47 42 47 33 30 30 30 A X A B B B C C C 64 64 56 50 50 42 36 36 36 21 21 A A A B B B C C C 69 64 57 57 57 43 36 34 36 26 21 26 X C C A A A A A 64 64 54 54 54 43 37 37 37 21 21 X X X C A A A A A 69 62 54 62 54 43 37 34 37 27 27 27 X C C A A A A A the 2x11x25 box: 65 65 55 51 51 51 38 32 38 22 22 22 X 2 2 2 2 2 2 2 2 2 2 2 2 65 59 59 59 45 45 38 38 38 24 24 X X X 2 2 2 2 2 2 2 2 2 2 2 65 65 55 52 45 52 39 32 39 23 24 23 X 2 2 2 2 2 2 2 2 2 2 2 2 66 60 66 60 45 45 39 39 39 24 24 A A A 2 2 2 2 2 2 2 2 2 2 2 67 60 60 60 46 46 46 33 29 29 29 A X A 2 2 2 2 2 2 2 2 2 2 2 68 68 56 53 53 42 40 40 40 25 25 X X X 2 2 2 2 2 2 2 2 2 2 2 67 68 56 53 47 47 47 33 30 25 30 A X A 2 2 2 2 2 2 2 2 2 2 2 68 68 56 53 53 42 41 41 41 25 25 A A A 2 2 2 2 2 2 2 2 2 2 2 69 61 57 61 57 43 41 34 41 26 26 26 X 2 2 2 2 2 2 2 2 2 2 2 2 69 61 61 61 48 48 48 34 31 31 31 X X X 2 2 2 2 2 2 2 2 2 2 2 69 62 62 62 48 43 48 34 31 27 31 27 X 2 2 2 2 2 2 2 2 2 2 2 2 63 63 55 51 44 51 44 32 28 22 28 22 X 2 2 2 2 2 2 2 2 2 2 2 2 63 59 55 59 44 44 44 32 28 28 28 X X X 2 2 2 2 2 2 2 2 2 2 2 63 63 55 52 52 52 35 32 35 23 23 23 X 2 2 2 2 2 2 2 2 2 2 2 2 66 66 66 49 49 49 35 35 35 20 20 A A A 2 2 2 2 2 2 2 2 2 2 2 67 58 58 49 46 49 46 33 29 20 29 A X A 2 2 2 2 2 2 2 2 2 2 2 67 58 56 50 50 42 40 33 40 20 20 X X X 2 2 2 2 2 2 2 2 2 2 2 67 58 58 50 47 42 47 33 30 30 30 A X A 2 2 2 2 2 2 2 2 2 2 2 64 64 56 50 50 42 36 36 36 21 21 A A A 2 2 2 2 2 2 2 2 2 2 2 69 64 57 57 57 43 36 34 36 26 21 26 X 2 2 2 2 2 2 2 2 2 2 2 2 64 64 54 54 54 43 37 37 37 21 21 X X X 2 2 2 2 2 2 2 2 2 2 2 69 62 54 62 54 43 37 34 37 27 27 27 X 2 2 2 2 2 2 2 2 2 2 2 2 the 3x3x10 box: A A A x x x A A A A A A A x y x y A A A A A A A y y y A A A with A equal 15 15 10 10 . 14 11 11 . 14 12 12 15 13 13 10 . 14 13 11 . 13 13 12 15 15 10 10 . 14 11 11 . 14 12 12 the 3x3x10 box: 27 25 21 21 X1 10 10 27 25 22 22 X1 11 11 28 28 22 19 19 12 12 29 29 22 22 17 13 13 30 30 23 23 17 14 14 27 25 21 X1 X1 X1 10 26 26 26 X1 X1 X1 11 26 28 26 19 15 15 12 29 24 24 24 17 15 13 30 24 23 24 15 15 14 27 25 21 21 X1 10 10 27 25 20 20 X1 11 11 28 28 20 19 19 12 12 29 29 20 20 17 13 13 30 30 23 23 17 14 14 27 25 21 21 16 11 11 a symmetric one 27 25 22 22 16 12 12 28 28 22 19 19 13 13 29 29 22 22 17 14 10 30 30 23 23 17 14 10 27 25 21 18 16 18 11 26 26 26 18 18 18 12 26 28 26 19 15 13 15 29 24 24 24 15 15 15 30 24 23 24 17 14 10 27 25 21 21 16 11 11 27 25 20 20 16 12 12 28 28 20 19 19 13 13 29 29 20 20 17 14 10 30 30 23 23 17 14 10 the 4x4x5 box: Stack two 2x4x5 boxes (with X and U) such that the Xs match. The double X can be replaced by two Us. the 4x5x5 box: unique upto X-symmetry 27 22 18 18 18 27 22 X 14 14 27 X X X 11 28 28 X 15 11 29 29 29 15 11 27 22 18 12 18 26 26 X 12 14 27 X X X 11 28 21 X 13 13 29 21 29 15 11 24 22 24 12 10 26 22 Y 14 14 25 Y Y Y 10 28 28 Y 15 13 25 21 17 15 17 24 24 24 12 10 26 26 Y 12 10 25 Y Y Y 10 25 21 Y 13 13 25 21 17 17 17 the 5x5x11 box: 2 solutions upto X-symmetry after placing A1 and A2 A2 A2 A2 50 45 50 45 31 31 20 20 A2 A2 A2 52 45 45 45 33 33 21 21 A2 A2 A2 51 51 42 42 32 33 32 21 A2 A2 A2 52 46 46 46 33 33 21 21 A2 A2 A2 53 46 53 46 34 34 22 22 A2 A2 A2 50 50 50 37 29 31 29 20 A2 A2 A2 52 44 44 37 29 29 29 17 A2 A2 A2 52 51 44 42 32 32 32 17 A2 A2 A2 52 44 44 38 30 30 30 17 A2 A2 A2 53 53 53 38 30 34 30 22 A1 A1 57 48 57 48 37 31 31 20 20 A1 A1 58 48 48 48 36 36 27 27 17 A1 A1 58 51 51 42 42 36 27 19 19 A1 A1 58 49 49 49 36 36 27 27 17 A1 A1 59 49 59 49 38 34 34 22 22 A1 A1 57 57 57 40 37 40 25 25 25 A1 A1 58 X1 43 43 37 X2 25 18 25 A1 A1 X1 X1 X1 43 X2 X2 X2 18 19 A1 A1 58 X1 43 43 38 X2 26 18 26 A1 A1 59 59 59 41 38 41 26 26 26 A1 A1 54 54 54 40 40 40 23 23 23 A1 A1 54 X1 54 39 39 X2 23 18 23 A1 A1 X1 X1 X1 39 X2 X2 X2 19 19 A1 A1 55 X1 55 39 39 X2 24 18 24 A1 A1 55 55 55 41 41 41 24 24 24 . 49 47 49 47 31 27 31 27 . 51 47 47 47 33 27 27 27 . 50 50 41 41 32 32 22 22 . 51 48 48 48 33 28 28 28 . 52 48 52 48 34 28 34 28 . 49 49 49 38 31 31 31 17 . 51 46 46 38 33 26 26 17 . 51 50 46 41 33 32 26 22 . 51 46 46 39 33 26 26 18 . 52 52 52 39 34 34 34 18 57 57 42 42 36 36 24 20 20 58 58 43 43 38 29 24 29 17 58 50 50 41 41 32 32 22 22 58 58 44 44 39 30 25 30 18 59 59 45 45 37 37 25 21 21 54 57 54 42 38 36 24 20 17 54 54 54 43 38 29 29 29 17 55 55 55 40 40 40 23 23 23 56 56 56 44 39 30 30 30 18 56 59 56 45 39 37 25 21 18 57 57 42 42 36 36 24 20 20 53 53 43 43 35 35 24 19 19 55 53 55 40 35 40 23 19 23 53 53 44 44 35 35 25 19 19 59 59 45 45 37 37 25 21 21 the plane: Planar figure must be filled with the double u. 1 1 1 2 1 2 1 2 2 2 The plane can be tiled a a a a a a a c c c a a a c c c c b b b c c c b b b b d d d b b b d d d d a a a d d d a a a a c c c a a a c c c c c c c a a a a a a a a a a b b b b b b b b b b a a a a a a a a a a b b b b b b b b b b