Question 331949
The number of one side paintings on an nxnxn cube is 

{{{6(n-2)^2}}}

In this case, we are told that n=4. So we have:

{{{6(4-2)^2=24}}}

There are 24 little cubies with only one side painted.

The number with two sides painted is {{{12(n-2)}}}

{{{12(4-2)=24}}}

The number with three sides painted is always 8. They are on the corners and that is true regardless of the cube size. 

Those remaining are the ones in the center with no paint. That is

{{{(n-2)^3}}}

{{{(4-2)^3=8}}}

So, we have 24+24+8+8=64...as it should be.

Add up the formulas:

{{{8+(n-2)^3+6(n-2)^2+12(n-2)}}}

A pattern. A binomial. Add these up and it reduces all the way down to

{{{n^3}}}

Just as we expected. 4^3=64