Understanding the Problem: A Cube Painted Red and Cut into 64 Smaller Cubes

Imagine a scenario where a large cube is painted red on all its faces and then cut into 64 smaller cubes, each measuring 1x1x1 units. This problem involves understanding how many of these smaller cubes have exactly 6, 5, 4, 3, 2, 1, and 0 red faces. The objective is to break down the configuration and count the cubes appropriately.

Total Cubes

The original cube, now consisting of 64 smaller cubes, implies that it is a 4x4x4 cube. This can be determined mathematically by recognizing that (4^3 64).

Cubes with Different Numbers of Red Faces

6 Red Faces

Cubes with all 6 red faces would be located at the corners of the larger cube. However, since the cube is divided into smaller units and each small cube is 1x1x1, none of the small cubes can have all 6 faces painted without being at a corner. Therefore:
Cubes with 6 red faces: 0

5 Red Faces

Cubes with 5 red faces are located at the center of each face of the larger cube. In a 4x4x4 cube, each face has 1 such small cube in the center. Since a cube has 6 faces, there are:
Cubes with 5 red faces: 6

4 Red Faces

Cubes with 4 red faces are located at the edges of the larger cube, excluding the corners. Each edge of a 4x4x4 cube has 2 such small cubes. Since a cube has 12 edges, the total count is:
Cubes with 4 red faces: 12 times 2 24

3 Red Faces

Cubes with 3 red faces are the ones at the corners of the larger cube. A cube has 8 corners, thus:
Cubes with 3 red faces: 8

2 Red Faces

Cubes with 2 red faces are located on the edges of the cube, excluding the corners and the centers of each face. Each edge has 2 such cubes. Therefore for 12 edges, the total count is:
Cubes with 2 red faces: 12 times 2 24

1 Red Face

Cubes with 1 red face are located on the faces of the larger cube, excluding the edges and corners. Each of the 6 faces of the cube has 4x4 16 smaller cubes. Subtracting the edges 4 on each edge, minus the corners gives 8 cubes per face. Hence, for 6 faces, there are:
Cubes with 1 red face: 6 times 8 48

0 Red Faces

Cubes with 0 red faces are the internal cubes that do not touch any outer face. These form a 2x2x2 cube in the center, giving:
Cubes with 0 red faces: 2 times 2 times 2 8

Summary of Counts

Here is a summary of the counts for each number of red faces:
- 6 red faces: 0
- 5 red faces: 6
- 4 red faces: 24
- 3 red faces: 8
- 2 red faces: 24
- 1 red face: 48
- 0 red faces: 8

Conclusion

This problem not only involves understanding the spatial configuration but also aids in the development of visual and logical reasoning skills. It is a classic example of a geometric puzzle that can be approached through systematic analysis and counting techniques.