If you try and draw a 3D representation of a cube, then draw lines on each side the divide the edges equally into 3, you will see that the cube contains 27 smaller cubes consisting of three layers of 9 cubes.
Now consider the top square layer. The botton sides of that layer are hidden by the middle layer (ie not painted). The geometric centre cube of that layer only can only have 1 side that is painted. The corner cubes have 3 sides that are painted. The remaining cubes (ie the centre cubes of each of the 4 edges of the square have two sides the are painted. Therefore this layer has 4 cubes with 2 sides that are painted.
Now consider the bottom square layer. The same rational gives 4 cubes with 2 sides that are painted.
Now consider the middle layer. Both the top and bottom sides are hiddle from view (ie not painted). The geometric centre cube can have no sides painted as all edges are hidden from view. The middle cubes on each of the four edges can only have 1 side painted but each of the 4 corner cubes will have two sides painted.
Therefore the total number of cubes with just two sides painted is 4 from the top layer, 4 from the middle layer and 4 from the bottom layer giving a total of 16 cubes out of the total of 27 cubes.
