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Paint it black (Posted on 2007-04-02) Difficulty: 3 of 5
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    
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Not likely | Comment 2 of 9 |

Of the 27 cubes, 8 are corners, 12 are edges, 6 are centers and one is unpainted.

After painting when they are rebuilt we first need to consider whether the positions are correct.  There are (27 C 8) ways of getting the corners right, if they are then there are (19 C 12) ways of getting the edges then (7 C 6) ways of getting the centers.  Total probability so far = 8!*12!*6!/27!

Even if the positions are correct the orientations of the cubes must also be correct.  A cube has 24 different orentations.  For corner cubes 3 are correct (1/8 prob.)  For edges 2 are correct (1/12) and for centers 4 are correct (1/6).  Total probability here: (1/8)^8*(1/12)^12*(1/6)^6

The grand total would then be the product of these two which simplifies to
1 / 5465062811999459151238583897240371200


  Posted by Jer on 2007-04-02 12:24:25
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