Red Faces of Painted Cubes

For each of the six directions (up, down, north, south, east, west), let $I_d$ indicate that the face of the chosen unit cube pointing in direction $d$ is red. The number of red faces is $T=\sum_d I_d$.

The upward face of a unit cube is red exactly when that cube lies in the top layer of the big cube. The top layer contains $10^2=100$ of the $1000$ unit cubes, so

By symmetry the same holds for each of the six directions. By linearity of expectation,

Equivalently, summing red faces over all unit cubes counts each painted unit square once. That total is the painted area $6\cdot 10^2=600$. Dividing by the $1000$ cubes gives the mean.

Alternative. Count the cubes by type.

• The $8$ corner cubes have $3$ red faces each.
• The edge cubes number $12\cdot 8=96$ and have $2$ red faces each.
• The face-centre cubes number $6\cdot 8^2=384$ and have $1$ red face each.

The total is $24+192+384=600$, which again gives $600/1000=3/5$.

Generalisation. For an $n\times n\times n$ cube, the expected number of red faces is $6n^2/n^3=6/n$.