Step 1: When a 3×3×3 cube is painted on all outer faces and then cut into 27 unit cubes, the small cubes fall into four types: corner cubes (3 faces painted), edge cubes (2 faces painted), face-centre cubes (1 face painted), and the one cube at the very centre (0 faces painted). Step 2: A cube painted on only one face is the small cube sitting exactly at the centre of each face of the large cube. Step 3: The large cube has 6 faces, and each contributes exactly one such centre cube. So there are 6 small cubes painted on only one face.