Santiago's Challenge

There's an old riddle...

An explorer leaves on a hike, traveling south for 10km, and then turning east and traveling another 10km, where he shoots a bear. Finally, the explorer turns north and hikes 10km to return to his exact starting point. What color was the bear he shot?

It's not too hard to realize that the bear must have been white, because the only place on earth where that particular hike is possible has the North Pole as its starting point.



In a recent post the NYS ESPRIT earth science teachers' listserve, Greg Wagner of Island Trees Middle School, Levittown, NY, described how his curious and clever student, Santiago, realized that if we leave out the bear part (and maybe substitute a bird), there is a different solution to the problem:
If you start from some distance north of the South Pole, and hike southward 10km, you can turn to the east and circle the South Pole on a parallel of latitude whose circumference is 10km, and then return north to the starting point (of course it's possible to circle the pole twice, or three times, etc. on parallels that are 5km, 3.33km, etc. in circumference).

So here is what we call "Santiago's Challenge":

Determine the exact latitude of the explorer's starting point near the South Pole, and the latitude of the point where he should turn to the east. As you think about it, you'll discover that there are several assumptions you have to make. Describe how you arrived at your solution, and what assumptions you made and why, and send them to Greg, Steve Kluge, Bryce Hand, and Charles Burrows. We'll send along any interesting replies we get.