Question 62
Let $\mathcal{C}$ be a cube of side length $2$. We color each of the faces of $\mathcal{C}$ blue, then subdivide it into $2^{3}=8$ unit cubes. We then randomly rearrange these cubes (possibly with rotation) to form a new 3-dimensional cube.
What is the probability that its exterior is still completely blue?
What is the probability that its exterior is still completely blue?