South, East, North, and Home from Uncountably Many Places
The North Pole really is a solution, so the trap is only the words "and nowhere else". A mile north of the parallel whose whole lap measures 1/n of a mile, the eastward mile is n exact revolutions, which puts a starting circle 1.159 miles from the South Pole for one lap, 1.080 for two, 1.053 for three. Every point of every circle works, so the honest count is uncountable rather than infinite.