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A large white cube is painted red, and then cut into 27 identical smaller cubes. These smaller cubes are shuffled randomly. A blind man (who also cannot feel the paint) reassembles the small cubes into a large one. What is this probability that the outside of this large cube is completely red?

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    Did you try anything? What is your plan to approach the problem? – 2017-01-20

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These "complex" probability questions should always be addressed in a similar manner. Decompose them in smaller subproblems and solve them separately.

This problem could be decomposed in finding these probabilities:

  • probability that the inner center is in its place
  • probability that the $8$ corner pieces are at the corner positions
  • probability that each corner piece is correctly orientated
  • probability that the $6$ center pieces are at the center positions
  • probability that each center piece is correctly orientated
  • probability that the $12$ edge pieces are at the edge positions
  • probability that each edge piece is correctly orientated

If you find those, you can multiply all of them and get the final result.

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Hint: The cube has a $3 \times 3$ cube. The set of "types" is $\mathrm{S} = \{\mathrm{center, corner, edge, core}\}$. The cardinality of each of those types is:

  • $| \mathrm{center}|= 6$,
  • $| \mathrm{corner}|= 8$,
  • $| \mathrm{edge}|= 12$, and
  • $| \mathrm{core}|= 1$.
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    actually |edge| = 12 or else they would not add up to the $27$ pieces – 2017-01-20
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    @RSerrao Thanks. Corrected. – 2017-01-20