Like an open box turned upside down, this cube is painted on its top and its four sides but not underneath. After cutting along the lines, how many small cubes have exactly two painted faces?

Two painted faces: along the 4 top edges, the 3 non-corner cubes each (12). Along the 4 upright edges, every cube except the top corner, 4 each (16), because the bottom cube of an upright edge has only its two side faces painted. The bottom edges give one painted face only. Total $12+16=28$. (36 is the answer for a cube painted all over.)