A wooden cube is painted red on all six faces and is then cut into 27 identical smaller cubes. How many of the small cubes have exactly two faces painted red?

Cutting into 27 equal cubes means 3 cubes along each edge. A small cube shows exactly two painted faces when it sits along an edge of the big cube but not at a corner. Each of the 12 edges has 3 − 2 = 1 such cube, so the count is $12\times 1=12$. (The 8 corner cubes show three faces and the 6 face-centre cubes show one.)