Edited by Richard Walker, Monday, 13 Jan 2025, 12:59
To reiterate this question, imagine we are rowing on a round pond 200 m across when suddenly fog descends and visibility is effectively zero. You are lost and unfortunately the fog fell too quickly for you to have any idea of your location relative to the pond's edge. What is the shortest distance you can row to guarantee reaching the edge of the pond and what path should you follow?
To begin with, any straight path of length 200 m must be a diameter of the pond. Therefore, if we choose an direction at random (we do not know where the shore lies relative to our location so one direction is as good as another) and travel in a straight line, we are certain to reach the bank after at most 200 m. So this is am escape path.
It's natural to wonder if some ingenious non-linear path might guarantee hitting the pond's edge in a shorter distance, but we can show this is impossible.
Here's the proof.
Suppose we can guarantee reach the shore in a distance of less than 200 m by following some path starting at A and ending at M and let M be the point half-way along the path, so the path length for M to A and to B are both less than 100 m.
Consider an arbitrary point X somewhere along the path. The distance along the path from M to X must be less than 100 m and the straight line distance from M to X must therefore also be less than 100 m. Since X was an arbitrary point it follows that every point on the path is less than 100 m from M and we can draw a circle with centre M and radius less than 100 m.
But this is smaller than the pond, which has a radius of 100 m! So it's possible for the whole of AB to lie within the interior of the pond and it therefore cannot form a guaranteed escape path.
My solution to "Lost in the Fog" Question 1
To reiterate this question, imagine we are rowing on a round pond 200 m across when suddenly fog descends and visibility is effectively zero. You are lost and unfortunately the fog fell too quickly for you to have any idea of your location relative to the pond's edge. What is the shortest distance you can row to guarantee reaching the edge of the pond and what path should you follow?
To begin with, any straight path of length 200 m must be a diameter of the pond. Therefore, if we choose an direction at random (we do not know where the shore lies relative to our location so one direction is as good as another) and travel in a straight line, we are certain to reach the bank after at most 200 m. So this is am escape path.
It's natural to wonder if some ingenious non-linear path might guarantee hitting the pond's edge in a shorter distance, but we can show this is impossible.
Here's the proof.
Suppose we can guarantee reach the shore in a distance of less than 200 m by following some path starting at A and ending at M and let M be the point half-way along the path, so the path length for M to A and to B are both less than 100 m.
Consider an arbitrary point X somewhere along the path. The distance along the path from M to X must be less than 100 m and the straight line distance from M to X must therefore also be less than 100 m. Since X was an arbitrary point it follows that every point on the path is less than 100 m from M and we can draw a circle with centre M and radius less than 100 m.
But this is smaller than the pond, which has a radius of 100 m! So it's possible for the whole of AB to lie within the interior of the pond and it therefore cannot form a guaranteed escape path.