Each cube below is made up of smaller cubes, but the large cubes are not solid. They had tunnels through them.
• The first cube originally had 27 small cubes, but the tunnel removed 3 cubes.
• The second cube originally had 64 small cubes, but two straight tunnels, 4 cubes deep, removed some cubes.
• The third cube, which originally had 125 small cubes, has 3 straight tunnels, five cubes deep from face to face.
The outside surfaces of these cube constructions have been painted including inside the tunnels and on the bottom. For each construction, how many small cubes have paint on 4 faces? 3 faces? 2 faces? 1 face? 0 face?
To find out the number of small cubes that have paint on a certain number of faces, we need to understand how the tunnels affect the cube constructions.
1. First cube:
The first cube originally had 27 small cubes, but the tunnel removed 3 cubes. Since the tunnels don't affect the faces of the remaining cubes, all 24 cubes that are not part of the tunnel will have paint on 3 faces. The remaining 3 cubes that were part of the tunnel will have paint on 2 faces since they are now exposed to the sides and the bottom.
Summary:
- Small cubes with paint on 4 faces: 0
- Small cubes with paint on 3 faces: 24
- Small cubes with paint on 2 faces: 3
- Small cubes with paint on 1 face: 0
- Small cubes with paint on 0 faces: 0
2. Second cube:
The second cube originally had 64 small cubes, but two straight tunnels, 4 cubes deep, removed some cubes. The tunnels will remove a total of 8 cubes (2 tunnels x 4 cubes each). Therefore, the remaining 56 cubes will have paint on their outside faces.
Summary:
- Small cubes with paint on 4 faces: 0
- Small cubes with paint on 3 faces: 56
- Small cubes with paint on 2 faces: 0
- Small cubes with paint on 1 face: 0
- Small cubes with paint on 0 faces: 8 (cubes removed by tunnels)
3. Third cube:
The third cube originally had 125 small cubes and has 3 straight tunnels, five cubes deep from face to face. Each tunnel will remove a total of 25 cubes (5 cubes deep x 5 cubes wide). Therefore, the remaining 100 cubes will have paint on their outside faces.
Summary:
- Small cubes with paint on 4 faces: 0
- Small cubes with paint on 3 faces: 100
- Small cubes with paint on 2 faces: 0
- Small cubes with paint on 1 face: 0
- Small cubes with paint on 0 faces: 25 (cubes removed by tunnels)
In conclusion:
1. First cube:
- Small cubes with paint on 4 faces: 0
- Small cubes with paint on 3 faces: 24
- Small cubes with paint on 2 faces: 3
- Small cubes with paint on 1 face: 0
- Small cubes with paint on 0 faces: 0
2. Second cube:
- Small cubes with paint on 4 faces: 0
- Small cubes with paint on 3 faces: 56
- Small cubes with paint on 2 faces: 0
- Small cubes with paint on 1 face: 0
- Small cubes with paint on 0 faces: 8
3. Third cube:
- Small cubes with paint on 4 faces: 0
- Small cubes with paint on 3 faces: 100
- Small cubes with paint on 2 faces: 0
- Small cubes with paint on 1 face: 0
- Small cubes with paint on 0 faces: 25