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?

(In reply to re: solution by matt)

Overall, if all the faces were distinguishable, there are 24 ways of orienting a cube when each face must be parallel or perpendicular to given planes (the larger cube): 6 * 4.  That is, there are 6 choices for what goes on top, and then, for each of those, there are 4 choices of which of the adjacent faces is to be toward a particular direction.

In regard to the specifics: For edges, two adjacent faces are black. Only one of the 12 edges of this cube is shared by these two black faces, and it is that one that must be aligned with the edge of the larger cube. It doesn't matter if the two black faces are reversed; that's why it's not 1/24, but rather 1/12.  Another way to look at it is that 1/3 of the faces will do for the first black position, then only one of the four faces adjacent faces will do for rotation into the position of the other outer face.

For the vertices, there are 3 black faces of the little cube. These all adjoin only one of the 8 vertices of that little cube, and it's that vertex which must go into the corresponding corner of the big cube.

For the face center of course, you need only choose the correct face. of the little cube. You can spin it into any of the 4 orientations along the axis perpendicular to that face and it doesn't matter.

Posted by Charlie on 2007-04-04 16:13:44