prime boxes of (41) list complete Torsten Sillke, BI, 10.04.93 1 1 1 2 2x5x 6p, 10p, 12, 14p, 15p, 16, 17p, 18, 19p, ... {14..19}+6n 3x5x 4p, 5p, 6p, 7p, ... {4..7}+4n 4x5x 3p, 4p, 5p, ... {3..5}+3n 5x5x 3p, 4p, 5p, ... {3..5}+3n 6x5x 2p, 3p, ... {2..3}+2n 2x10x 4p, 5p, 6, 7p, ... {4..7}+4n 3x10x 3p, 4, 5, ... {3..5}+3n 2x15x 4p, 5p, 6, 7p, ... {4..7}+4n 3x15x 3p, 4, 5, ... {3..5}+3n 2x2x2x2x 10p, ... 10n Complete list of prime boxes (# = 19): 2x4x{10,15} 2x5x{6,10,14,15,17,19} 2x7x{10,15} 3x3x{10,15} 3x4x5 3x5x{5,6,7} 4x4x5 4x5x5 5x5x5 Impossible: 2x2...2x5u (with u odd), 2x2x2xN, 2x2xZ, 2x3xN 2x5x{4, 5, 7, 8, 9, 11, 13} 3x3x5 prime boxes of (41/42) (with reflection) 2x5: 3p, 4p, 5p, ... {3..5}+3n 3x5: 2p, 3p, ... {2..3}+2n 2x2x2x2: 10p, ... 10n Impossible: 2x2...2x5u (with u odd) 2x2xZ 2x2x2xN Proof: impossibility of 2x2...2x5u (with u odd) Look at the 1-dimensional projection. It will be shown that you can't cover n*(1,1,...,1) of length 5u with (...,0,3,1,1,0,...), (...,0,3,1,1,0...). Reduce the the number of variables further: Use the projection: (a,b,c,b,c,b,c,...,c,b,a). This gives the linear system: [ 3 1 0 0 ] _ [a] [ 1 1 4 1 ] * X = [b] [ 1 3 1 4 ] [c] It will be shown that this system has no integral solution if 3 is no divisor of n. But n is a power of 2, so there is no tiling. To show that the system has no integral solution, try to solve it modulo 3. Then the system reduces to: [ 0 1 ] [a] [ 1 1 ] * Y = [b] (modulo 3) [ 1 0 ] [c] To show, that (a,b,c) is not in the span of (1,1,0) and (0,1,1) you are allowed to cancel the factor n, as it is not divisiable by 3. It remain three different (a,b,c) to test, set u = 1..3 which give (2,2,1), (2,1,0) and (2,0,2). In each case is a+c <> b and the system therefor insolvable. QED Proof: impossibility of 2x2xN and 2x2x2xN Look at the 1-dimensional projection. It will be shown that you can't cover (4,4,4,...) and (8,8,8,...). To cover the first place, you can use (3,1,1,...) or (1,1,3,...) only. But in both cases the third place is covered if the first is coverd. But the second place is disconnected from the rest and still uncoverd. QED Annotations: without reflection The 3x4x5 box: symmetric 20 17 13 13 13 20 17 14 11 11 20 20 15 16 11 21 21 21 16 11 19 17 17 17 13 19 18 14 11 13 21 20 15 10 12 21 16 16 16 12 19 19 19 14 10 18 18 14 14 10 18 15 15 10 10 18 15 12 12 12 The 3x5x5 box: unique 21 21 14 14 14 22 21 15 15 15 23 23 23 11 11 23 18 24 12 12 24 24 24 13 12 21 17 15 16 14 22 17 15 16 14 19 19 19 10 11 23 18 24 12 11 20 20 20 13 11 21 17 17 17 10 22 16 16 16 10 22 22 19 10 10 20 18 19 12 13 20 18 18 13 13 The 4x5x5 box: 28 28 19 14 14 28 20 20 20 14 29 21 21 22 13 29 22 22 22 13 29 23 23 23 13 26 20 19 14 10 28 20 18 11 11 29 29 21 22 11 27 23 21 12 11 27 23 21 13 13 26 19 19 14 10 28 19 18 11 16 24 24 16 16 16 27 27 27 12 15 25 25 17 17 17 26 18 18 10 10 26 26 18 10 16 25 24 15 15 15 25 24 17 12 15 25 24 17 12 12 The 5x5x5 cube: 13 solutions with piece 10 fixed in this position. 33 33 21 21 21 33 29 21 15 15 34 32 22 20 15 34 25 22 20 15 34 34 22 20 19 31 31 31 14 14 33 29 21 15 14 32 32 22 22 14 32 25 19 19 19 32 34 20 20 19 29 29 31 14 11 33 29 31 10 13 30 10 10 10 13 30 25 18 24 13 30 25 25 24 12 26 23 23 23 11 26 23 16 10 17 30 30 17 17 17 27 27 18 13 13 28 24 24 24 12 26 26 26 11 11 27 23 16 11 17 27 28 16 16 16 27 28 18 12 12 28 28 18 18 12 The 5x5x5 cube: 41 solutions with piece 10 fixed in this position. 33 33 33 16 16 33 27 30 17 16 34 32 23 17 16 34 32 23 17 18 34 32 31 18 18 30 30 30 16 13 33 27 30 17 17 34 34 23 23 23 31 31 31 15 18 32 32 31 15 15 27 27 21 19 13 28 27 10 19 11 10 10 10 19 12 29 24 22 22 18 29 29 22 14 15 25 25 21 13 13 28 25 10 13 11 26 25 19 19 12 26 24 20 20 20 26 29 22 14 15 25 28 21 11 11 28 28 21 21 11 26 26 20 12 12 24 24 20 12 14 24 29 22 14 14 The 5x5x5 cube: 131 solutions with piece 10 fixed in this position. 33 33 20 20 20 33 31 21 21 21 34 31 21 14 14 34 31 24 15 15 34 24 24 16 15 30 30 19 14 20 33 30 19 14 20 34 34 21 14 16 31 31 24 15 16 32 32 22 16 16 30 23 19 19 19 33 23 23 17 10 32 29 10 10 10 32 29 24 15 13 32 29 22 13 13 30 25 18 18 18 26 26 23 17 10 27 27 27 12 11 28 22 22 12 13 29 29 22 12 12 25 25 17 17 18 25 26 23 17 18 25 26 27 11 11 28 26 27 11 13 28 28 28 11 12 The 5x5x5 cube: 30 solutions with piece 10 fixed in this position. 33 25 31 15 14 33 25 31 15 12 34 25 25 15 15 34 26 26 22 21 34 26 22 22 21 33 33 33 14 14 31 31 31 14 12 34 34 25 14 15 32 32 32 22 11 32 26 21 21 21 29 24 24 12 12 30 27 24 13 12 30 30 10 13 13 10 10 10 22 11 32 26 20 20 20 29 24 19 17 16 27 27 19 17 16 27 30 10 17 13 27 23 20 11 11 28 28 20 11 18 29 24 19 19 19 29 29 16 16 16 28 30 17 17 13 28 23 18 18 18 28 23 23 23 18 The 5x5x5 cube: 45 solutions with piece 10 fixed in this position. 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 Annotations: The 2x2x2x2x10 Box (one half) 5 5 2 4 2 4 2 4 7 8 2 4 7 8 9 9 6 6 2 4 b c b b d e f g b c a a d e f g b c c c d e 5 6 1 3 1 3 1 3 5 6 1 3 7 8 9 a 5 6 1 3 7 8 9 a d d f f 7 8 9 a . . . . e e g g