Painted Blocks Problem & Solution

We have lots of cool problems that our instructor Zandra put together for our Resources page! The one below is from the American Mathematics Competitions and also appeared in our January newsletter. The solution is immediately below it, so don't scroll down too far if you want to think about the problem before seeing the solution! 


Cubes have six faces, so a cube that has exactly four red faces will have exactly two blue faces.  These faces will have remained blue because they were shielded by being pressed against another cube initially, in the figure.  So it is the cubes that are in contact with exactly two other cubes in this figure that we need to count – the cubes with two neighbors.

The figure is very symmetric, such that there are really only three cases to judge:  
1) The four raised blocks in the structure’s corners are each in contact with only the one neighboring block beneath each.  
2) The four lower-level corner blocks are each in contact with two blocks on the lower level and the one atop it, three neighbors altogether.
3) And the remaining six blocks that go between the two-block towers are each sandwiched between two neighbors.  Therefore, these six blocks will have exactly two blue faces.

Therefore, the solution is six.