When a cube is cut into \(4 \times 4 \times 4 = 64\) smaller cubes:

• Cubes with 3 painted faces: Only corner cubes = 8
• Cubes with 2 painted faces: Edge cubes (not corners). Each edge has \(4 - 2 = 2\) such cubes. A cube has 12 edges. So = \(12 \times 2 = 24\)
• Cubes with 1 painted face: Face cubes (not on edges). Each face has \(2 \times 2 = 4\) such cubes. 6 faces = \(6 \times 4 = 24\)
• Cubes with 0 painted faces: Inner cubes = \(2 \times 2 \times 2 = 8\)

Answer: 24 cubes have paint on exactly two faces.