To analyze the scenario of coloring a 4-colored cube composed of \( 343 \) smaller cubes (which can be interpreted as a cube with dimensions \( 7 \times 7 \times 7 \), since \( 7^3 = 343 \)), we can denote the colors as \( C_1, C_2, C_3, \) and \( C_4 \).
The requirement is that no two consecutive cubes on the sides can be of the same color. This interprets into a grid coloring problem with specific adjacency restrictions.
Step 1: Understanding the Cube
In a \( 7 \times 7 \times 7 \) cube, there are \( 7^3 = 343 \) total smaller cubes. Each smaller cube can be colored one of 4 colors \( C_1, C_2, C_3, \) or \( C_4 \).
Step 2: Adjacency Constraint
The adjacency constraint ("no two consecutive cubes of the same color") implies that for each axis-oriented layer of cubes, the coloring must alternate in such a way that adjacent cubes in any direction (left-right, front-back, top-bottom) cannot share the same color.
Step 3: Checkerboard Pattern
One effective way to maximize the number of cubes of each color, while adhering to the adjacency constraint, is through utilizing a checkerboard-like pattern.
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Split each face of the cube in a checkerboard fashion. In a \( 7 \times 7 \) layout, let’s color it such that:
- Cubes at positions \( (i, j) \) where \( i + j \) is even can be one set of colors, and those where \( i + j \) is odd can form another set of colors.
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Since each face is \( 7 \times 7 \), there will be:
- \( 25 \) cubes where \( i + j \) is even (as \( \lceil \frac{49}{2} \rceil \)), and
- \( 24 \) cubes where \( i + j \) is odd (as \( \lfloor \frac{49}{2} \rfloor \)).
Step 4: Extending to 3D
By applying such a checkerboard pattern across all three dimensions, we can represent each of the four colors across distinct even or odd layers, ensuring no adjacent cubes in space share the same color.
- In total, we can assign cubes as follows:
- Color \( C_1 \): cubes at positions satisfying the first condition (even positions).
- Color \( C_2 \): cubes at odd but not adjacent positions of \( C_1 \).
- Colors \( C_3 \) and \( C_4 \): assigned in similar alternating patterns.
Conclusion
Given the arrangement across the entire 3D structure while meeting the constraints provided by the problem, the maximum count of smaller cubes of each color is \( 343/4 = 85.75\).
Since the number of smaller cubes must be a whole value (integers), the maximum number of smaller cubes of each color can be constrained to about \( 85 \) or \( 86 \) for different colors, depending on the exact coloring pattern used:
- Thus, the arrangement could yield either \( 86 \) cubes of two colors and \( 85 \) cubes of the other two colors, representing the optimal distribution.
Hence, the answer is that at most 86 smaller cubes of each color are possible, with adjustments made for the constraints (two colors having slightly more).