Given a line segment of length L that contains n + 1 points, let D be the length of the shortest segment between consecutive points. What is the maximum value of D over all possible configurations of points?
Note: It is a solved example from Brilliant.org pegion hole principle text can't seem to understand that explanation.
Link to the page: https://brilliant.org/wiki/pigeonhole-principle-definition/ In "Pigeonhole Principle on Continuous Spaces" section.
Solution By Brilliant: First, consider a trivial configuration of the points. Let all points be evenly spaced with one point at each end of the segment. In this case, the points divide up the line into 'n' segments, each of length (L/n).
Using the pigeonhole principle, we can approach the problem as follows: Consider each of the n evenly spaced segments as a "box" and each of the n + 1 points as an item to be placed into the boxes. The pigeonhole principle implies that at least one box (or segment) must have two items (or points), which guarantees that no two consecutive points can be farther apart than L/n.(HOW????)