Prime Packings of the Pentacube 35: Torsten Sillke, FRA 16.06.98 2 2' 1 1 Z*Z: 2, 4, 5, ... {4, 5}+2n open cases 3-dim: N*Z*n N*N*N N*N*Z Impossible: N*N*n Z*Z*3 Proof: impossibility of ZxZx3 (with reflections) Color cube (x,y,z) with z (modulo 2). This coloration makes a distinction between outer and inner cubes. The ratio of outer:inner cubes is 2:1. But each piece can have at most a ratio of 3:2. As the pieces have bounded size, no packing is possible. Proof: impossibility of n x N x N (with reflections) (Proof of H. Postl, June 1998) It will be shown that a bounded edge (corner to corner) can not be covered. It is sufficient to consider the nx2x2 subbox. We want to fill the edge-cubes (these are (x, 0, 0)) from left to right. We will see that there are only two different cases after the placing the first piece (with one exception). case 1: (y,z) x-1 x x+1 --------------- (0,0) x 0 0 (1,0) . . . (0,1) . . . (1,1) x x . case 2: (y,z) x-1 x x+1 x --------------- --- (0,0) x 0 0 0 (1,0) . x . and the reflection . (0,1) . . . x (1,1) x . . . Notation: x : filled cube . : don't care cube 0 : free cube Therefore the (0,0) row will be allways behind one of the three others. But then we never get a right corner. The exception mentioned above is that the first two pieces completly fill the first 2-cube but not more of the nx2x2 subbox. Then we must fill the remaining (n-2)x2x2 subbox. The critical case is if we have the iterated 2-cube case. Then try to fill the next nx2x2 subbox. Now we have additional cubes from the first nx2x2 subbox, that we must have case 1 or 2. Annotations: the 2*Z*Z plane: there are many solutions. building block: 2 2 2 2 2 2 2 2 2 2 building block strip: a a a a a b b a a a b b b c c a a b b b c c c d d b b c c c d d d e e c c d d d e e e d d e e e e e the 5*Z*Z plane: e e a a e f f a b b f g g b c c g h h c d d h d A e e E a a B f f F b b C g g G c c D h h H d d A A E E A B B E F F B C C F G G C D D G H H D H a e A a b E e f B b c F f g C c d G g h D d H h a a e e a b b e f f b c c f g g c d d g h h d h References: - H. Postl Pentacube 35 is not boxable. The 5*5*n boxes of the Q pseudo pentomino. letter from 8. June 1998