## The Returning Explorer: Answer

Posted 16th July 2017 by mikail

Yesterday we asked you about the returning explorer. If you haven’t read the question, make sure you read it first.

### The Obvious Answer

The obvious answer is of course that **the bear is white**. The explorer started from the north pole. He first walks a mile south, so now he is one mile away from the north pole. When starts walking west, he walks along a circle around the north pole. At each point along the circle, he is exactly one mile away from the north pole. When he finally walks north again, he walks along the radius of the circle back to its centre at the north pole. This means that the wild bear must be a polar bear.

### An Infinity of Not-So-Obvious Answers

If we think of other places that the explorer might have started, we find that he could have also started near the south pole. Of course, there are no polar bears at the south pole, only penguins.

To find out where exactly, we start by looking at the westward leg of the journey. If the explorer is near the south pole, then the walk due west prescribes a circle around the south pole. One of those circles will have a circumference of exactly one mile. So the mile long walk due west will return the explorer to the start of that leg. If we go north one mile from there, we find the starting point of the explorer. Now, because the westward leg could have started anywhere on the circle the possible starting points also prescribe a circle around the south pole. I will let you figure out the radius of that circle for yourselves.

We already have infinitely many starting points along a circle around the south pole. But we can find even more! Instead of walking around the south pole only once, the explorer could have walked around the south pole 2 or 3 times or more. If he walks twice around the pole, we find a circle with a circumference of half a mile. If he walks three times around the pole, the circle must have a circumference of one-third of a mile, and so on. For each of these circles, the possible starting points of the explorer is a circle with a radius that is one mile larger.

Can you calculate the radii of these circles? Post your answers in the comments.