A few hikers set off along a trail that stretches infinitely ahead. The trail is narrow, so no one can overtake. Each hiker walks at a distinct constant speed. When a faster hiker catches a slower one, they form a group and continue at the slower hiker's pace. Initially, three hikers are equally spaced along the trail.
Suppose the same three hikers began in a different order along the trail, but each kept their original starting speed. On average (averaging over all possible initial orderings), how many groups will remain after a long time?
How does the answer change if there are 4 hikers? 5 hikers? In general, what happens for hikers?
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PROMYS Europe (2026)
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