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Unpainting Black (Posted on 2017-07-14) Difficulty: 4 of 5
Like in Paint it Black, twenty-seven identical white cubes are assembled into a single cube; then the outside of that cube is painted black.

The cube is then disassembled and rebuilt randomly.

What is the probability that the outside of this cube is now completely white?

No Solution Yet Submitted by Brian Smith    
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re: Preliminary work & thoughts | Comment 2 of 5 |
(In reply to Preliminary work & thoughts by Jer)

I agree with the probabilities except for the edge type located now at a vertex.  I see this as 3/12 or the same 1/4 as a vertex piece now located at an edge:

When an edge piece is at a vertex there is one edge on that piece that has two black faces adjacent to it. This edge must be at one of 3 out of the 12 possible positions for this edge-- the three edge positions that are internal to the larger cube.

Now the matrix is symmetric with respect to the major diagonal; there must be a reason for this, also.

BTW, while it's convenient to call the possible set of states a matrix, it's a 4-dimensional array of 9x13x7x2. It's sparsely populated as at each stage the sum of the indices has a particular value: 27 at the start, then 19 after choosing the corner pieces, then 7 after choosing the edges, which then leaves only one. The array dimensions run of course from zero to 8, 12, 6 and 1 respectively.

Edited on July 14, 2017, 10:20 am
  Posted by Charlie on 2017-07-14 10:13:03

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