Update Sept 3: Solution (to original version) is here
Update 7/31: The solution to this puzzle requires not much math at all, if you think about it the right way. I’m tempted to only award beer if someone gets the more complex version (at the end of post–that one also requires only minimal math), but I’ll stick to my word instead.
I went running in the park today, and thought of the following puzzle: suppose you are going running in the park, and that there is one loop-shaped running path which you will run along for a total of 30 minutes. You have a friend who will be running along the loop for exactly the same 30 minutes as you (i.e. you start and end at the same time), and at exactly the same speed as you.
You have no idea where your friend will start his or her run along the loop, nor do you know what direction your friend will run in (clockwise our counterclockwise) but you do know that your friend will run in a single direction for the entire 30 minutes.
You want to maximize the probability that you will run into your friend while running. All that you have control over is which direction you are running in at any point in time along your run. So one possibile course of action is to run in one direction for the entire 30 minutes (either clockwise or counter). Another is to run in one direction for a certain period of time, and then (assuming you haven’t met) switch directions and finish the run in the other direction. Or you can switch directions multiple times, each after specified period of time.
So, what is the optimal course of action in order to maximize the probability of meeting? If there are multiple courses of action that give the equivalent, maximal, probability, what are they? The answer may depend on how much of the loop you cover in 30 minutes (but assume that you cover less than
half the full loop–otherwise you can clearly guarantee a meeting)*. Assume that you are very nearsighted, so you only see your friend when you are actually crossing paths. And remember that you and your friend travel at the same speed, and you have no control over, or knowledge of, where you start in relation to your friend, or the direction of his/her run.
I will buy, for the first person who can solve this puzzle (i.e. give an answer and prove it), 2 beers at a bar.
For those of you who don’t like getting too deep into the weeds with puzzles like this: can you make any statements generally about the strategy you should employ (no beer awarded for such statements)?
More complex version: Modify the puzzle by specifying that you and your friend don’t travel at the same speed, but rather that the ratio of your speed to your friend’s speed is X.
*edited 7/31. if solution is provided based on original language, it’s ok as far as the beer award is concerned.