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