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 Paint it black (Posted on 2007-04-02)
Twenty-seven identical white cubes are assembled into a single cube, and 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 again completely black?

 See The Solution Submitted by Federico Kereki No Rating

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 re: solution | Comment 5 of 9 |
(In reply to solution by Charlie)

The 12 edge pieces must be oriented to the edge with the painted pair of faces is along the edge of the big cube, with probability, for all 12:

(1/12)^12

Why is it 1/12? - I think that there are only 6 ways to orient a cube.  All pieces are interchangeable with their same type - edges can be used in any edge spot, middles in any middle

For the faces:

(1/6)^6

and for the vertices:

(1/8)^8

Why is it 1/8 - I think that there are only 6 ways to orient a cube.  All pieces are interchangeable with their same type - edges can be used in any edge spot, middles in any middle, with only 1 orientation of the 6 working to complete the pattern.

Except the chewy center, which has no orientation preference and only one way to choose....

In all the probability is

12! * 6! * 8! / (27! * 12^12 * 6^6 * 8^8)

I get 12!*8!*6!*1/( 27! * 6^26)

can someone else settle this?

Cheers.

 Posted by matt on 2007-04-04 14:09:17

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