
SOMA News 
19 Sep 2001
EMail. 
On 14. Sep. 2001 Nick Dickens. Albuquerque, New Mexico, USA. (CO: Pat Bateman <horrorkid@excite.com>) asked me an interesting question about coloring of a SOMA cube.
Here is the question, and two answers.
Question:
Is it possible to form a cube,
Paint the outside of the cube,
Take it apart,
And then form a cube again, but have no paint visible?
Answer: by Courtney McFarren
NO.
There are 8 corners in the 3x3x3 cube.
Each piece can contribute, at most:
pieces #1, 4, 5, 6 & 7: 1 corner each
pieces #2 & 3: 2 corners each
Which gives us a maximum of 9 corners, which is more
than the required 8.
But piece #3 has a strange property 
it can either donate 2 corners, or no corners,
but it is IMPOSSIBLE for it to donate just ONE corner.
If piece #3 donates NO corners, then the maximum number
of corners left is only 7, not enough to form the 3x3x3 cube.
(This was explained in the newsletter, called "Winning Ways").
Therefore, piece #3 must ALWAYS donate 2 corners, which
means it must ALWAYS have its spine along one of the edges
of the 3x3x3 cube. Because the exposed spine is painted in
one solution, then the painted spine will have to be exposed
for all the other 239 solutions.
Answer: by Bob Allen
"NO". The 2, 3, and 4 pieces are all three cubes long in
one dimension, so their "ends" must touch the outside of
any 3x3 cube that can possibly be built. They would always
be painted, and always show.