Final answer:
The problem is solved by understanding that the averaging process across 2000 points on a circle will result in all points converging to the same value due to the constraints of each point being an average of its neighbors.
Step-by-step explanation:
The problem given is a classic mathematics puzzle involving a circle and assigning values to points on that circle. Specifically, each point on the circle has a number that equals the average of the numbers of its two nearest neighbors. This situation creates a system of linear equations where the value at each point depends on the values of its neighbors.
To understand why all numbers must be equal, consider two adjacent points, A and B. Let's say point A has a value of x, and point B has a value of y. By the puzzle's rules, point B's value is the average of point A and another point C, which is next to point B. When the entire circle is filled in this manner, each point is constrained by its neighbors, leading to a system that stabilizes only when all values are the same.
If you attempted to have different values, the iterative process of averaging would continue to propagate changes around the circle until all inconsistencies are ironed out, resulting in all points reaching the same value. Thus by repeated application of the averaging rule, it can be shown that the numerical values must converge to a common value for all 2000 points on the circle.