Spring 2008 — 10. Two Hikers
Two hikers, Alice and Bob, hike the same trail for two days in succession. They both start at the same starting spot at 9 am. They both end up at the same finishing point at 3 pm. On both days, Alice is either at the same spot or ahead of Bob on the trail. Each day, at some point, Alice reaches a maximum distance of 200 yards ahead of Bob.
Solution
Jared Latiolais earned five points for the first correct answer:
Yes, it is always the case that, for some time, at that time on both days, the distance between Alice and bob is the same.
Consider a graph of how far Alice is ahead of Bob (d, in yards), as a function of time. Because Alice is either at the same spot or ahead of Bob, and because the maximum distance between the two is 200 yards, we know that d is always greater than or equal to 0 and always less than or equal to 200. The only times that occur twice during the hike are those between 9:00 AM and 3:00 PM. We can cut out one of the 9:00 AM to 3:00 PM and 0 yards to 200 too yards rectangles and superimpose it on the other. If the two segments of the distance curve intersect then we have a situation at that time where the two hikers are the same distance apart on both days. Since d = 0 at the beginning and end of the hike, the two segments will be contiguous curves, each with an endpoint at the bottom corners of the rectangle (the 200 yard x 6 hour section of possible points on the graph) and the other endpoint on the opposite vertical side of the rectangle. Regardless of where the second endpo! ints are and regardless of the shape of the curves, the two segments will intersect at at least one point.
I also received a correct answer from Paul Ottaway.