A cube painted blue all over is cut into 1000 equal small cubes. How many of the small cubes have exactly one blue face?

1000 small cubes means 10 along each edge. On each face, the cubes with one painted face form the inner $8\times 8=64$ squares. Six faces: $6\times 64=384$. (486 uses $9\times 9$ per face; 512 is the unpainted core $8^3.$)