Let's analyze the problem step-by-step:

Problem Restatement

• A solid cube has side length 12 cm.
• It is painted on its faces:
  • 2 adjacent sides painted green,
  • 2 other adjacent sides painted yellow,
  • 2 opposite sides painted blue.
• The cube is then cut into smaller cubes of side length 4 cm.
• We need to find how many of these smaller cubes have no face painted.

Step 1: Number of smaller cubes

• The large cube side = 12 cm.
• Smaller cube side = 4 cm.
• Number of smaller cubes along each edge = 12 / 4 = 3.
• Total smaller cubes = 3 × 3 × 3 = 27.

Step 2: Understanding the painting

• The large cube has 6 faces.
• Painting scheme:
  • 2 adjacent faces green,
  • 2 other adjacent faces yellow,
  • 2 opposite faces blue.

Since the cube has 6 faces, these painting instructions cover all faces:

• Let's label the faces for clarity:
  • Face 1 and Face 2: adjacent, painted green.
  • Face 3 and Face 4: adjacent (and different from Face 1 and 2), painted yellow.
  • Face 5 and Face 6: opposite faces, painted blue.

Step 3: Cubes with no painted faces

• Smaller cubes with no painted face are those completely inside the large cube, i.e., not touching any face of the large cube.
• Since the large cube is 3 × 3 × 3 smaller cubes, the cubes that have no painted faces are those that are not on any outer layer.
• The outer layer cubes are those on the faces, edges, or corners.
• The inner cubes are those in the "middle" layer.

Step 4: Counting inner cubes

• Since the cube is 3 × 3 × 3:
  • Outer layers are at positions 1 and 3 along each dimension.
  • The inner layer is at position 2 along each dimension.
• So, the inner cubes are at coordinates (2, 2, 2) only.
• Number of inner cubes = 1 × 1 × 1 = 1.

Final answer:

Only 1 smaller cube has no face painted. If you want, I can also explain how many cubes have one, two, or three painted faces!