A $10 \times 10 \times 10$-inch wooden cube is painted red on the outside and then cut into its constituent 1 -inch cubes. How many of these small cubes have at least one red face?
. There are 8 corner cubelets, each with three red faces. Each of the twelve edges has a row of 8 edge cubelets, each with two red faces. The cubelets with only one red face form $8 \times 8$ squares in the interior of each of the six faces. So the total count is

$$
8+12 \cdot 8+6 \cdot 8 \cdot 8=8+96+384=488 \text {. }
$$