To determine the number of small cubes with paint on each number of faces, we need to consider the effects of the tunnels for each cube construction.
For the first cube:
- The total number of small cubes is originally 27.
- However, 3 small cubes were removed due to the tunnel.
- Therefore, there are 24 small cubes remaining.
- Each small cube has 0 faces painted since the tunnels also removed paint from the inside.
For the second cube:
- The total number of small cubes is originally 64.
- Two straight tunnels, 4 cubes deep, were created.
- This means that a 4x4x4 cube in the center has been removed, leaving a 4x4x4 cube frame.
- The total number of small cubes remaining is (4x4x4) - (4x4) = 64 - 16 = 48.
- Each small cube on the outside surface of the frame has 3 painted faces.
- Each small cube on the inside surface of the frame has 2 painted faces.
- Each small cube on the edge of the frame (excluding corners) has 1 painted face.
- Each corner cube has 0 painted faces.
For the third cube:
- The total number of small cubes is originally 125.
- Three straight tunnels, five cubes deep, were created.
- This means that a 5x5x5 cube in the center has been removed, leaving a 5x5x5 cube frame.
- The total number of small cubes remaining is (5x5x5) - (5x5) = 125 - 25 = 100.
- Each small cube on the outside surface of the frame has 3 painted faces.
- Each small cube on the inside surface of the frame has 2 painted faces.
- Each small cube on the edge of the frame (excluding corners) has 1 painted face.
- Each corner cube has 0 painted faces.
Summary of the number of small cubes with paint on each number of faces:
1st Cube:
- 24 small cubes with 0 faces painted.
2nd Cube:
- 8 corner cubes with 0 faces painted.
- 16 edge cubes with 1 face painted.
- 24 inside surface cubes with 2 faces painted.
- 48 outside surface cubes with 3 faces painted.
3rd Cube:
- 8 corner cubes with 0 faces painted.
- 36 edge cubes with 1 face painted.
- 54 inside surface cubes with 2 faces painted.
- 100 outside surface cubes with 3 faces painted.
Note: It is important to double-check the cube configurations and tunnel positions to ensure accurate results.