1. The Returning Explorer

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AN OLD RIDDLE runs as follows. An explorer walks one mile due south, turns and
walks one mile due east, turns again and walks one mile due north. He finds
himself back where he started. He shoots a bear. What color is the bear? The
time-honored answer is: “White,” because the explorer must have started at the
North Pole. But not long ago someone made the discovery that the North Pole is
not the only starting point that satisfies the given conditions! Can you think of
any other spot on the globe from which one could walk a mile south, a mile east,
a mile north and find himself back at his original location?


Is there any other point on the globe, besides the North Pole, from which you
could walk a mile south, a mile east, and a mile north and find yourself back at
the starting point? Yes indeed; not just one point but an infinite number of them!
You could start from any point on a circle drawn around the South Pole at a
distance slightly more than 1 + 1/2π miles (about 1.16 miles) from the Pole—the
distance is “slightly more” to take into account the curvature of the earth. After
walking a mile south, your next walk of one mile east will take you on a
complete circle around the Pole, and the walk one mile north from there will
then return you to the starting point. Thus your starting point could be any one of
the infinite number of points on the circle with a radius of about 1.16 miles from
the South Pole. But this is not all. You could also start at points closer to the
Pole, so that the walk east would carry you just twice around the Pole, or three
times, and so on.
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