A cube is painted on five faces, every face except the bottom one, and then cut along the lines. How many small cubes have no paint at all?

An unpainted cube must keep away from the top and the four sides, but it may sit on the bottom. Away from the sides leaves a 2 by 2 column in the middle; away from the top leaves 3 of its 4 layers. So $2\times 2\times 3=12$. (8 treats the bottom as painted.)