To determine how many different cubes of various colors could be visible from a single point in space on a colored cube made up of 343 small 1x1x1 cubes, let's first analyze the structure of the overall cube.
The cube consists of \( 343 = 7 \times 7 \times 7 \) small cubes. The entire cube itself is 7 cubes along each edge.
When viewed from a vantage point, we can typically see three faces of the cube (the top face and two adjacent side faces). We need to ensure that no two adjacent small cubes on these visible faces have the same color.
Given that there are 4 different colors available, we can assign colors to the small cubes while adhering to the restriction of not having adjacent cubes of the same color.
Let's set up a coloring scheme based on the structure:
- Each face of the cube is made up of \( 7 \times 7 = 49 \) small cubes.
- Only the outer layer of cubes is visible on these three faces. The three visible faces share some edges and corners at their intersections.
Counting visible cubes
Since we are looking at three faces:
- The top face has 49 cubes.
- The two side faces also have 49 cubes each, but they share 7 cubes along the edge with the top face and share 7 cubes on the vertical edge between them.
From the above:
- Total cubes from three faces without accounting for overlaps: \( 49 + 49 + 49 = 147 \)
- Overlap from the shared edge between the top and each side face (7 cubes each): \( -7 - 7 \)
- Overlap from the corner cube (counted in both side faces and on top): \( +1 \)
So, the total unique visible small cubes:
\[ 147 - 14 + 1 = 134 \text{ cubes} \]
Conclusion on the maximum number of visible colors
To maximize the number of visible colors while respecting the adjacent color restriction, we can use a checkerboard-like pattern on the faces. Given that there are 4 colors and color assignments can be chosen judiciously, you can easily keep up with the requirement of different colors on adjacent cubes.
Based on the checkerboard coloring on such patterns, essentially you can effectively show all 4 colors visible from a single viewing point while adhering to the adjacent color restriction, especially since you're viewing three faces.
Answer
Thus, at most 4 cubes of different colors can be visible from a single point in space while satisfying the color adjacency condition.