there are 7 identical spheres each colored on the inside with one color in more than 4 of them. When 2 spheres of the same color touch each other they both glow in color. Using as few comparisons as possible, find a single spehere that has the majority color. How many comparisons do you need?
To solve this problem efficiently, we can follow a strategy known as "tournament method" or "pairwise comparison".
First, divide the 7 spheres into three groups - A, B, and C - containing 2, 2, and 3 spheres respectively. Each group will have at least one sphere of the majority color.
Now, we compare Group A and Group B to find the majority color. Since each group has 2 spheres, it will take at most 1 comparison to determine the majority color within each group.
Case 1: If the colors of the spheres in Group A and Group B are different, we proceed to the next step.
Since there is a sphere of the majority color in Group A and one in Group B, we can eliminate one sphere from each group that is not of the majority color. Now, we have 1 sphere in Group A, 1 sphere in Group B, and 3 spheres in Group C.
Case 2: If the colors of the spheres in Group A and Group B are the same, we can conclude that the majority color is the color of the spheres in these groups. In this case, the problem is already solved with just one comparison.
Next, we have 1 sphere in Group A, 1 sphere in Group B, and 3 spheres in Group C. We compare the spheres in Group A, B, and C as follows:
- Compare the sphere in Group A with any sphere in Group B (1 comparison).
- Compare the remaining sphere in Group B with any sphere in Group C (1 comparison).
If the majority color is found during these comparisons, we can determine the sphere with the majority color. Otherwise, it means that the remaining sphere in Group A has the majority color.
In the worst-case scenario, the majority color is not found until the final comparison. Therefore, the total number of comparisons needed is 1 (Group A vs. Group B) + 1 (Group B vs. Group C) + 1 (final comparison) = 3 comparisons.
Therefore, you can find the single sphere with the majority color using a minimum of 3 comparisons.