SOMA News 19 Sep 2001 E-Mail.

# Coloring a SOMA cube.

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.

Edited by Thorleif Bundgaard.

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