Thorleif's SOMA page

Well, I thought that this was a problem that would stand for long, but already 2 days later ...
Bob Nungester sendt me a mail. Very thoroughly describing the thoughts going into explaining this one.
Mathematicians use the sign ' ⊂ ' meaning that a 'set' is a 'subset of' another 'set'. So this is a subset of SOMA ;-)

Bob's explanation

I read the question about painting two opposite sides of a SOMA cube and then hiding the painted
squares in another cube solution. At first there seemed to be too many possibilities to try,
but after some analysis it became much simpler and resulted in three painted solutions that
can be hidden. In SOMATYPE notation these are U0h, U3d, and U7d.
They can all be hidden in any of the 37 members of the "A" family of solutions, or else in their reflection.

/SOMA-One Solution
/222/766/776
/152/155/736
/144/445/333
↓ ↓
/222/744/441
/256/771/731
/556/566/333

Possible Solutions - Top/Bottom surface are painted
U0h U3d U7d
2 2 2 6 2 2 4 2 2
1 5 2 6 5 2 4 5 2
1 4 4 1 1 2 1 1 2

7 6 6 6 6 7 7 6 6
1 5 5 4 5 5 4 5 5
4 4 5 4 1 5 4 1 5

7 7 6 4 7 7 7 7 6
7 3 6 4 3 7 7 3 6
3 3 3 3 3 3 3 3 3

The first of 37 Solutions - use the one that hide the piece 2 black squares

A0a A0aR
2 2 2 2 2 2
5 6 2 2 5 6
5 6 6 5 5 6

4 4 7 7 4 4
1 7 7 7 7 1
5 5 6 5 6 6

1 4 4 4 4 1
1 3 7 7 3 1
3 3 3 3 3 3

Here's the methodology used to find these:

This analysis uses the concepts of Central and Deficient from the SOMAP, as described and illustrated
in the Winning Ways newsletter. Figure 3 of that newsletter shows all possible positions for each piece.
These positions can have the cube rotated and reflected in any of 48 ways, but the same faces are always exposed.
Any cube solution can be rotated to the normalized position, with pice 3 at the front bottom with its fourth
cubelet also on the bottom. The file SomapNormalized.txt shows all 240 solutions in this configuration.

First, note that only the top and bottom faces can be painted to possibly lead to a hidden solution.
As noted by Courtney McFarren and Bob Allen, the ends of piece 3 can never be hidden so the left and right
sides of any cube cannot be painted. Similarly, the back side of piece 3 cannot be hidden so the front and
back side of any cube connot be painted. This leaves only the top and bottom faces to be analyzed.

Pieces 5 and 6 can have one, two or three squares painted based on their position in the top and bottom faces.
It's best to minimize their painted squares, so having only one or two painted squares will make a painted
solution more likely to support hiding these squares. This isn't a necessary requirement, but we only need to
find one solution to prove it's possible. A good position that usually results in one painted square on piece 5 or 6
is the Central and Deficient position. These configurations in SOMATYPE notation are the "R" and "U" families
of solutions, so start with only these. Now eliminate any solution where the other 5 or 6 piece has three
squares painted.
In addition, any solution with piece 2 or 4 spanning from the top to bottom of the cube can be eliminated,
because these two ends are always visible in any cube solution. With piece 5 or 6 Central and Deficient
all other pieces must be in the Normal position. The Normal position of piece 2 always has its 3-square
back spine visible, so any painted solution with this back spine painted can be eliminated as well.

With all these rules it can be seen that all "R" family solutions are eliminated and only the three "U" family
members listed above are left. Now on to the possible hidden solutions.

Piece 7, like all the others, is in the Normal position in all the painted solutions, so one of the faces with
three squares is painted. The only way to hide these is the only other possible placement of Central and Deficient.
This placement is the "A" family of solutions, so only these (or their reflection) can result in a hidden solution.
Similar to the pained solutions, all pieces except 7 are in the Normal position.

Piece 1 can have any one of three faces painted, but it can be placed either of two ways in any given position.
One or the other of these placements will always hide the painted face, so any of the "A" family of solutions
hides this piece. Piece 2 can only have one of its sides painted since the other faces were eliminated in
possible painted solutions. This face will be hidden in any solution or in its reflection.
Piece 3 always has its painted face hidden by simply flipping it over.
Piece 4 has two painted squares in each of the three "U" family solutions being tested, and that face can be
hidden in any cube with it in the Normal position by rotating it to one of its two possible orientations.
Pieces 5 and 6 have one or two painted squares and those faces can also be hidden by rotating the piece
to one of its two orientations. Piece 7 has its three painted squares hidden in all "A" family solutions.
This means that any of the 37 "A" family solutions, or their reflection, can hide all three of the "U" family
solutions listed.

The restriction of having piece 5 or 6 Central and Deficient wasn't proven to be a necessary condition,
so there could be other possible painted solutions that can be hidden.