Thursday, March 4, 2010

Painted cubes

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 Answers:

Consider separately the four types of cubes upon disassembly:

1. 8 cubes with three faces painted black;
2. 12 cubes with two black faces;
3. 6 cubes with one black face;
4. 1 completely white cube.

Then count the number of ways in which all cubes of each type may be correctly positioned and oriented.

The probability that the outside of the reassembled cube is completely black is 1/5465062811999459151238583897240371200 = 1/(256 · 322 · 52 · 7 · 11 · 132 · 17 · 19 · 23) is approximately equal to 1.83 × 10−37.

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