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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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Some Thoughts Preliminary work & thoughts | Comment 1 of 5
This problem is much harder than the original.  To be completely black, every type of piece (Center, Face, Edge, Vertex) had to be in the correct position and then in the correct orientation.  Then we could just multiply the probabilities.  Here any piece can be in any position as long as the orientation is correct.  

Here is a probability matrix for each type of piece to be in the correct orientation for each position:

type\position	c	f	e	v
center		1	1	1	1
face		1	5/6	2/3	1/2
edge		1	2/3	5/12	5/24
vertex		1	1/2	1/4	1/8

The problem is these 27 cubelets can be distributed among the positions in 27! ways.  (Well, technically 27!/1!/6!/12!/8! but each of these has its own probability of occurrence.)  

My idea from here is completely impractical:  For each of these ways create a new matrix of the number of each type in each position.  These would become exponents on the probabilities above and then multiply the 16 entries gives the probability of that arrangement working.  Do that for every possible distribution of cubelets and add these probabilities to get the final answer.



  Posted by Jer on 2017-07-14 09:35:58
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