CAT 2016 Slot 2 DILR: Painted Cube Cutting

Only cubes at corners can have 3 painted faces (two side faces + one more). The 4 bottom corners have only 2 painted faces each (two sides; the bottom is unpainted and they don't touch the top). The 4 top corners have 2 sides + the top painted = 3 painted faces. So exactly 4 unit cubes have 3 painted faces — the four top corners.

Unpainted cubes are those not touching any painted face: not touching either side face, not touching the top, but they DO include the layer touching the unpainted bottom. This forms a 4×4×5 block (the inner 4×4 cross-section, 5 layers up from the bottom, since the topmost layer touches the painted top). Count = 4×4×5 = 80.

By exhaustive check of all 216 unit cubes by (x,y,z) position: faces painted = 0 → 80 cubes, 1 → 96, 2 → 36, 3 → 4 (total 216). The 36 cubes with exactly 2 painted faces lie along the edges where two painted faces meet (the edges between the 4 side faces and each other, and between the side faces and the top), plus the 4 bottom corners (which get 2 painted side faces but no top or bottom paint).

Cubes with exactly 1 painted face sit at the centre of a painted face, away from all its edges. Each of the 4 side faces (6×6) contributes a 4×5 interior block (4 columns away from the two vertical side-edges, 5 rows since only the top edge of that face is shared with another painted face — the bottom edge is safe as the bottom is unpainted): 4×5 = 20 cubes per side face, ×4 = 80. The top face (6×6) contributes a 4×4 interior block (away from all 4 edges, each shared with a painted side face): 4×4 = 16. Total = 80 + 16 = 96.