Question

a women starts at a point P on the earths surface and walks 1 mi south, then 1 mi east, then 1 mi north, and finds herself back at P, the starting point. Describe all points P for which this is possible

Answer 1

Let point P be the center of the North pole. 

Any direction she moves 1 mi, she is walking 1 mi South.

1 mile is a very small distance but we can still think of the path as a little curved because of the shape of the Earth, but she is just moving in a straight path. 

Then the woman moves East, and then back North.