## Thursday, February 11, 2010

### Not Exactly Rubiks Cube

Twenty-seven identical white cubes are assembled into a single cube, the outside of which is painted black.  The cube is then disassembled and the smaller cubes thoroughly shuffled in a bag.  A blindfolded man (who cannot feel the paint) reassembles the pieces into a cube.  What is the probability that the outside of this cube is completely black?

1. The probability that every type of cube is in the right place (0, 1, 2, & 3 black sides) is

12!8!6!1!/(27!)

Then you must ensure that each cube is oriented correctly and each cube has 24 possible orientations. So the probability of that is

((3/24)^8)*((2/24)^12)*((4/24)^6)

Multiply the results of those together and you get the probability to be

1/5465062811999459151238583897240371200
or
1.83x10^-37

2. Woohoo, I got the same answer as Jay...

(8!12!6!)/(27!) * (1/12)^12 * (1/8)^8 * (1/6)^6

I'm just too slow with it.

3. Did you think it was going to be so small when you started thinking about it?

4. 4/27, maybe?

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