All 8 corner cubes are removed from a $5\times 5\times 5$ cube built of small cubes. The remaining solid is then painted all over, including the hollows left by the corners, and separated into its small cubes. How many small cubes have exactly two painted faces?

Without the corners each edge has 3 cubes. The two end cubes of every edge now also show the face that touched the removed corner, so they have 3 painted faces. Only the middle cube of each edge has exactly 2: $12\times 1=12$. (36 forgets the faces exposed by the hollows; 24 is the number of 3-face cubes.)