The 8 corner cubes are taken out of a $4\times 4\times 4$ cube built from small cubes. The solid that is left is painted all over, inside the corner gaps too. How many small cubes now have exactly three painted faces?

Before the corners were removed, the 24 edge cubes (2 on each of 12 edges) each had two outer faces. Removing a corner exposes one more face of each of the three edge cubes touching it. Every edge cube touches exactly one corner, so all 24 now have three painted faces; nothing else does.