Consider a point about 1.16 miles (JJJJ to be precise) from the South Pole (there are many to choose from). The first leg of the journey takes him (or her - it works just as well for women) to within about 0.16 mile of the pole. The second leg takes him in a circle around the pole, back to the point that he has just left, and then in the third leg he retraces the first, in reverse, ending up where he started.
If he starts a little closer (about 1.08 miles) the same applies, but the second leg takes him twice around the pole, or, closer still (a little more than 1.05 miles) and he circumnavigates the pole 3 times.
And so on . . .
So, far from there being just one point which satisfies the conditions, there are an infinite number of points on each of an infinite number of circles. How many is that altogether? Now, there's a real problem ...
Head of IT Malvern Girls' College 15 Avenue Road Malvern Worcestershire