Torsten Sillke, 2000-10-24 x x x x x x x 3x7x 3p, 4p, 5p, ... {3,4,5}+3n 4x7x 3p, 4p, 5p, ... {3,4,5}+3n 5x7x 3p, 4p, 5p, ... {3,4,5}+3n Complete list of prime boxes (# = 6): {3, 4, 5}x{3, 4, 5}x7 Impossible: everything is possible. Annotations: . 1 1 1 block X has two solutions 1 1 1 1 1 1 1 1 1 1 1 . 3 3 block P has two solutions 3 3 3 3 3 . 1 1 1 . block Q . 1 1 1 . 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 . block Z 1 1 1 . . 1 1 . . 1 1 1 . 1 1 1 The 3x3x7 Box: P1 P1 15 15 11 11 11 the right block A is unique P1 P1 15 15 12 12 12 P1 P1 15 15 15 10 10 P1 P1 14 14 11 11 11 P1 P1 14 14 12 12 12 P1 P1 14 14 14 10 10 P1 P1 P1 13 13 13 11 P1 P1 P1 13 13 13 12 P1 P1 P1 13 10 10 10 P1 y y 18 18 18 14 the right block B is unique P1 y y 18 18 18 14 P1 y y y 15 18 14 P1 P1 P1 17 17 17 14 P1 P1 P1 17 17 17 14 P1 P1 P1 17 15 15 15 P1 P1 P1 16 16 16 14 P1 P1 P1 16 16 16 14 P1 P1 P1 16 15 15 15 The 3x4x7 Box: 17 17 17 12 this block C has a unique solution 17 17 13 13 17 17 13 13 The 3x4x7 is B plus C. 18 14 13 13 13 16 16 16 12 18 16 16 12 18 16 16 12 18 14 14 14 15 15 15 12 18 15 15 12 18 15 15 12 18 14 14 14 The 4x4x7 Box: 17 X1 X1 X1 this block D has a unique solution X1 X1 X1 X1 X1 X1 X1 X1 The 4x4x7 is two times D plus a X block X1 X1 X1 14 17 17 17 13 16 13 13 13 16 13 13 13 16 14 14 14 17 17 17 12 16 12 12 12 16 12 12 12 16 14 14 14 . . . . . . . . . . . . 16 . . . The 3x5x7 Box: 23 23 23 16 this block E has a unique solution 23 23 23 16 23 20 18 16 The 3x5x7 is B plus E plus a single piece. 24 20 18 18 18 24 20 18 18 18 22 22 22 16 22 22 22 16 22 20 17 17 24 20 17 17 24 20 17 17 17 21 21 21 16 21 21 21 16 24 20 21 19 24 19 19 19 24 19 19 19 23 23 23 16 this block F has a 3 solutions 23 23 23 16 24 20 23 16 The 3x5x7 is F plus A. 24 20 20 20 24 20 20 20 17 22 22 22 16 22 22 22 16 22 19 19 19 24 19 19 19 24 19 17 17 17 21 21 21 16 21 21 21 16 21 18 18 18 24 18 18 18 24 18 17 17 17 1 a a a b b b a a A A A b b 1 a a a b b b a a A A A b b 1 a c d b 2 3 a a a c b b b c c c d d d 3 B B B c c c C c c c d d d 3 B B B c c c C 1 a a a b b b E E E A A F F 1 a a a b b b E E E A A F F c c c a d 2 b E G G G F F F c c d d d 2 3 B B G G C C C c c d d d 2 3 B B G G C C C 1 a a a b b b E E E A A F F 1 a a a b b b E E E A A F F c c c a d 2 b E G G G F F F c c d d d 2 3 B B G G C C C c c d d d 2 3 B B G G C C C The 4x5x7 Box: 23 23 23 16 16 This block and its mirror immage 24 20 y y y give the 4x5x7 24 20 y y y 24 x x x y 23 23 23 16 16 24 20 y y y 24 20 y y y x x x x y 23 20 16 16 16 24 20 y y y 24 20 y y y x x x x y z z . . . z z . . . z z z . . x x x . . 10 10 10 B1 B1 This block and its mirror immage T1 T1 B1 B1 B1 give the 4x5x7 T1 T1 B1 B1 B1 T1 T1 B1 B1 B1 10 10 B1 B1 B1 T1 T1 B1 B1 B1 T1 T1 B1 B1 B1 T1 T1 B1 B1 B1 10 10 B1 B1 B1 T1 T1 B1 B1 B1 T1 T1 B1 B1 B1 T1 T1 B1 B1 B1 20 20 . . . 20 20 . . . 20 20 20 . . T1 T1 T1 . . The 5x5x7 Box: 10 10 10 B1 B1 Two times this block plus the Q gives the 5x5x7 box T1 T1 B1 B1 B1 T1 T1 B1 B1 B1 T1 T1 B1 B1 B1 T1 T1 T1 Z1 Z1 10 10 B1 B1 B1 T1 T1 B1 B1 B1 T1 T1 B1 B1 B1 T1 T1 B1 B1 B1 Z1 Z1 Z1 Z1 Z1 10 10 B1 B1 B1 T1 T1 B1 B1 B1 T1 T1 B1 B1 B1 T1 T1 B1 B1 B1 Z1 Z1 Z1 Z1 Z1 . . . . . . . . . . . . . . . . . . . . Z1 Z1 . . . The 4x8x7 Box (not prime): 3 3 3 2 3 3 3 2 times 2 2 2 2 2 2 2 + 7 times X = 4x8x7 2 2 2 2 2 2 2 2 2 2 3 2 2 2 a a A A A b b B B B A A C C B B B A A C C B c c c C C C a a A A A b b B B B A A C C B B B A A C C B c c c C C C a a a . b b b . . . . . . . . . . . . . . . . . c . . . -- mailto:Torsten.Sillke@uni-bielefeld.de http://www.mathematik.uni-bielefeld.de/~sillke/