prime boxes of P6                  Torsten Sillke

 x x
 x x x x

 Nx 2, 3, ... {2,3}+2n

 3x 4p, ..., 4n
 6x 2p, 4, 6, 8, 10, 11p, ... {10,11}+2n
 9x 4, 8, 10p, ... {8, 10}+4n
12x 2, 3, ... {2,3}+2n
15x 4, 6, ..., {4,6}+4n
18x 2, 4, 5p, ..., {4,5}+2n
21x 4, 6, ..., {4,6}+4n
(12,15,18,21)+12n 

2x3x 4, 6, ..., {4,6}+4n
3x3x 4, 8, 10p, ..., {8,10}+4n
4x3x 1p, ..., 4n
5x3x 4, 6p, ..., {4,6}+4n
6x3x 2, 4, 5p, ..., {4,5}+2n

Impossible:
 2x(6n+3)
 3x(4n+2)
 6x{5,7,9}
 2x...x2x3x...x3x(2n+1)
 3x...x3x6

 Impossible 2x...x2x3x...x3x(2n+1):    (there my be no axis of length 3)
 Proof: look at a line parallel to the last axis.
   Each piece will match it an even number of times.

 Impossible 3x...x3x6
 Proof: project on the last axis and color it: w w b b w w.
   The relation w:b = 2:1 that the only possible 1-dim 
   placings are 2 2 1 1 0 0 and 0 0 1 1 2 2. Modulo 2 this
   is 0 0 1 1 0 0 but the box has 1 1 1 1 1 1. So the ends
   cannot be made odd.

Annotations:

 3x4:
 1 1 1 1
 1 1 2 2
 2 2 2 2

 2x6:
 1 1 1 1 2 2
 1 1 2 2 2 2

 reptile:
 1 1 1 1
 1 1 2 2
 3 3 2 2 2 2 4 4
 3 3 3 3 4 4 4 4

 6x11:
 a a a a a a 1 1 a a a
 a a a a a a 1 1 a a a
 b b b r r r r 1 a a a
 b b b r r r r 1 a a a
 b b b r r r r r r r r
 b b b r r r r r r r r

 9x10:
 1 1 1 1 a a a a a a
 r r 1 1 a a a a a a
 r r b b b b c c c c
 r r b b b b c c c c
 r r b b b b c c c c
 r r r r a a a b b b
 r r r r a a a b b b
 r r r r a a a b b b
 r r r r a a a b b b

 18x5:
 a a a a c c c c 5 5 5 5 b b b b b b
 a a a a c c c c 5 5 4 4 b b b b b b
 a a a a c c c c 2 2 4 4 4 4 a a a a
 b b b b b b 1 1 2 2 2 2 3 3 a a a a
 b b b b b b 1 1 1 1 3 3 3 3 a a a a 

 3x3x10:
 a a a a 1 1 1 1 1 1   with {a, b, c} 1x3x4 boxes and {1} 3 1x2x6 boxes.
 2 2 2 2 1 1 1 1 1 1
 2 2 b b b b c c c c

 3x5x6:
 a a a a e f   with {a, b, c, d, e, f} 1x3x4 boxes.
 b b b b e f
 c c c c e f
 1 1 1 1 e f
 1 1 d d d d