Step 1: In a 3×3×3 cube, the maximum number of painted faces any small cube can have is 3, and these are exactly the small cubes sitting at the corners of the large cube. Step 2: A cube has 8 corners, so there are 8 small cubes with 3 painted faces, and none has more than 3. So 8 small cubes are painted on at least 3 faces.