Math Puzzle: The Great Escape
Level of Difficulty: Highschool
You are in the middle of a field surrounded by a circular fence of radius R. Your objective is to escape by reaching the fence. However, life isn’t so rosy. Patrolling the perimeter is a guard dog.
At every instant the dog tries to minimize the distance between the point nearest to you on the fence and him (however, the dog is confined to the perimeter):
The bad news is that, while your maximal speed is v, the dog’s maximal speed is 4v. This means that you can’t just make a straight dash for it (can you see why?). You must come up with an escape strategy. What will it be?
(Note: for the record, dogs make one of the most loyal and affectionate pets there are and if you ever come across an ill-tempered one, the owner is to blame.)
EDITED, 18/June/2009: a solution has been posted here. Hats off to our reader DU for his solution in the comments section.
First time reader, so I don’t know how much rigor you are looking for. Here’s what I do:
First, I run in any direction so that I’m more than (1-π/4)R (= ~.21R) from the center but less than .25R. At .25R, I would be able to exactly pass the dog if we both ran in circles, so at less than .25R I should be able to outrun him. So I keep running circularly until I’m directly across the center from him. Then I run straight out to the fence.
For simplicity, let’s say I ran out exactly (1-π/4)R in step 1. My remaining distance is then (π/4)R. His distance is πR. Since he runs 4x my speed, we tie. But I can actually have run out a little farther, up to .25R, so I could win.
A little confused and probably not optimal…
At .25R, I would be able to exactly pass the dog if we both ran in circles…
should read
At .25R, I would be able to exactly pace the dog if we both ran in circles…