Unpainted unit cubes inside a dipped 10-cube

A large $10\times10\times10$ cube is assembled from $1000$ unpainted unit cubes. The whole large cube is then dunked in paint. How many of the small unit cubes end up with no paint on any face?

$512$. Only cubes on the outer layer touch paint. Stripping that one-cube-thick shell from every side leaves an inner block of $8\times8\times8=512$ cubes, all untouched.