Question

Consider a 4 colored cube of 343 small cubes each of dimension 1x1x1

having one particular color. No two consective cubes on the other sides

can be of same color. 

(c) At most how many cubes of each color in diagonal would be visible from the same point.

Answer 1

To analyze the problem of the colored cube in question, we note that it consists of 343 smaller cubes, which suggests that the larger cube must have dimensions of \(7 \times 7 \times 7\) (since \(7^3 = 343\)).

The problem states that the cube is colored with 4 colors, and it includes the constraint that no two consecutive cubes on the other sides should have the same color. This suggests that the coloring is applied in a way such that it follows this rule consistently throughout the visible faces of the cube.

When observing the cube from a particular viewpoint, we can see:

1. **Three visible faces** of the cube will be exposed, allowing us to view the cubes from the front, top, and one side.
2. There are cubes along the diagonals of these three visible faces.

### Diagonal Visibility
For a \(7 \times 7\) cube visible from a corner, each visible face has a diagonal from one corner to the opposite corner:

- Each face has a diagonal that runs from one vertex to the opposite vertex.
- Each face will have 7 cubes along its diagonal.

### Cube Reflection
When you look at the diagonal of one face, cubes not only populate one diagonal but can also align with cubes on the other diagonals of the adjacent faces.

#### Counting Unique Diagonal Cubes
When considering how many cubes can take a specific color along the visible diagonals:
1. **Main Diagonal (from a corner view)**:
- Each face has diagonal cubes that are visible.
- For a \(7 \times 7\) cross-section, the bottom left to top right diagonal contains 7 cubes.

2. **Each of the three visible faces contributes its own diagonal**:
- There will be overlap of cubes where each visible diagonal may have a cube that is also part of the diagonals on adjacent faces (not double-counted).

3. **At most, 3 unique cubes from these diagonals** can be visible per color at any time if color constraints are maintained.

### Conclusion
Since no two consecutive cubes on the adjacent faces can share the same color, it is reasonably deduced:
- The maximum possible cubes of a particular color in a diagonal that are visible (considering different colored small cubes) is limited by two factors:
- Color distinctiveness among adjacent cubes visible.
- Total cubes available in a singular color along each diagonal from a vertex view.

Hence, from the diagonal viewing, we can conclude that **at most 3 unique cubes of each color in diagonal would be visible from the same point.**