Here is part (c).
Analyzing the Cube and the Viewing Point
Understanding the Problem:
We have a 7x7x7 cube with alternating colors on adjacent faces.
We want to maximize the number of cubes of a single color visible in a diagonal.
Key Points:
The maximum number of cubes visible in a diagonal depends on the orientation of the viewing point.
We need to find the optimal viewing point to maximize the number of visible cubes of a single color.
Determining the Maximum Visible Cubes
Optimal Viewing Point:
To see the maximum number of cubes of a single color in a diagonal, we need to position the viewing point directly in front of a corner of the cube. This will ensure that we can see the longest diagonal.
Visualizing: Imagine the cube with a corner facing you. You'll see a diagonal going from the top corner to the bottom corner of the opposite face
.
Counting the Cubes:
In a 7x7x7 cube, the longest diagonal consists of 7 cubes.
Since the colors alternate on adjacent faces, every other cube in this diagonal will be the same color.
Calculation:
Number of cubes in the diagonal = 7
Half of the cubes = 7/2 = 3.5
Rounding down, we get 3.
Therefore, the maximum number of cubes of each color visible in a diagonal from a single point is 3.
