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Wandering The Earth
Thanks John E

White bear.
: starting at the North Pole will return you to the North Pole after the 3 manoeuvres (south along a line of longitude, east along a line of latitude and north again along a line of longitude).

You can start anywhere on a line of latitude (which is a circle) which is 1 mile north of the point where the circumference of the earth is 1 mile.
You will see that this is very close to the South Pole.
So you start by travelling 1 mile south, 1 mile east - a circle which brings you back to the "1 mile south position" - then travel back 1 mile north.
The sketch is below
(when last discussed with a friend, we were talking in kilometres, so you have the option of changing the sketch or rewriting the question in kilometres).
The fact you can start anywhere on the circle of latitude gives you an infinite number of equal solutions.
Further, you can get another infinite set of solutions by moving towards the South Pole so that the "eastward mile" is 1 loop, 2 loops, 3 loops, etc., each of these solutions bringing you closer to the South Pole.
In the limit, of course, the starting point is within a hair of 1 mile north of the South Pole.

south pole sketch

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