Question

How many points are there on the globe where by walking one mile south, one mile east and one mile north you reach the place where you started?

Answer 1

In the Northern
hemisphere, there is
exactly one. The North
Pole. It is easy to see why
this is a solution.
In the Southern
hemisphere the solution is
a little harder to see. The
actual South Pole itself is
out as you can't walk
southward from that
point. However if you start
a certain latitude north
from that pole, you can
walk a mile southward to
arrive at a point in the
periphery of a latitudunal
circle that's exactly a mile
in circumference. Walking
a mile north would mean
traversing the same exact
path backward to the
start. Any point on that
starting latitude circle
satisfies the condition.
Now consider a point on a
latitude even closer to the
South Pole where walking
south will put you at the
periphery of a latitude
circle that's half a mile in
circumference. Walking a
mile east would take you
two times around to the
same point, so this is
clearly a solution as well.
Turns out you can find an
(countably) infinite
number of starting latitude
circles each with an
(uncountably) infinite
number of starting points
on them in the Southern
hemisphere.

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