4*4*4 cube!

A large white cube is painted red, and then cut into 64 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. Let P P denote the probability that the outside of this large cube is completely red?

What is the value of 1 0 85 × P 1.26 10^{85} \times P *1.26 ? (Answer to two decimal places)

This is 100% original!


The answer is 9.85.

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1 solution

Kenny Lau
Jul 22, 2015

PREREQUISITE:

  • There are four types of smaller cubes: corners, edges, faces, centres.
  • Corners have three faces adjacent to each other coloured red.
  • Edges have two faces adjacent to each other coloured red.
  • Faces have one face coloured red.
  • Centres have no face coloured red.
  • There are 8 8 corners, 24 24 edges, 24 24 faces, and 8 8 centres.

The total number of cases:

  • is the arrangement of the 64 64 smaller cubes so that all the types go to where they belong.
  • is 64 ! 8 ! 24 ! 24 ! 8 ! \dfrac{64!}{8!24!24!8!}
  • therefore the probability for now is 8 ! 24 ! 24 ! 8 ! 64 ! \dfrac{8!24!24!8!}{64!} .

Additional limitations:

  • each corner has 8 ways of rotating itself, where only 1 is correct.
  • each edge has 12 ways of rotating itself, where only 1 is correct.
  • each face has 6 ways of rotating itself, where only 1 is correct.
  • each centre has 1 way of rotating itself, where all is correct.

THEREFORE:

  • P = 8 ! 24 ! 24 ! 8 ! 64 ! 8 8 1 2 24 6 24 1 8 P = \dfrac{8!24!24!8!}{64!8^812^{24}6^{24}1^8}

P.S. I noticed that I calculated 9.83 \fbox{9.83} but the answer is 9.85 \fbox{9.85} .

The answer is 9.85 because the poster initially made false calculations in the probability and edited the question accordingly.

Vishnu Bhagyanath - 5 years, 10 months ago

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I would like to hear the mistakes that you claim that he made.

Kenny Lau - 5 years, 10 months ago

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