One Saturday morning at 10:00 you hop on your bike in Oberlin to ride up to the beach in Lorain, a distance of about 9 miles. You do not travel at a constant speed: you stop to rest, go fast and slow, and then, maybe have to fix a flat tire. You arrive at the beach at noon. You stay for the afternoon and then camp out there for the night. The next morning, again at 10:00, you head home for Oberlin, by exactly the same route. You generally travel faster this time, and you arrive some time shortly after 11:00.
Can it be that there is a point along your route that you passed at exactly the same time on both mornings? Explain your answer.
- There might be,
- There must be,
- There can’t be.
Solution
The answer is b: There must be. And there will only be one such point (unless you double back on one day or the other).
The same distance must be covered and unless there had been a change in the route, some point has to be passed at the same time. A clever way to solve this is to clone yourself. At the moment you begin your return ride on Sunday from Lorain, your clone executes a re-run of your ride from Oberlin. There will be a moment when the two of you will pass each other, and that is the point you passed at exactly the same time on both mornings.