One mile south

problem: how many places are there on the earth that one could walk one mile south, then one mile east, then one mile north and end up in the same spot? to be precise, let's assume the earth is a solid smooth sphere, so oceans and mountains and other such things do not exist. you can start at any point on the sphere and walk in any direction you like. think you've figured it out? i'll tell you now, there is more than one. in fact, there are more than two. also be advised that walking north from the north pole (or south from the south pole) is illogical and therefore does not enter into the problem. all normal assumptions about directions will be used.

there are no tricks involved with this question. it just forces you to really think about the problem to come up with all the solutions.

solution:

well the north pole is one such place.

then somewhere near the south pole such that when you walk one mile south you are at the point on the earth where the circumference is 1. that way when you walk 1 mile east, you end up back at the same point. and of course one mile north from there puts you back where you started. here is a drawing courtesy of jy. there may or may not be such a place in the northern hemisphere where walking a mile south puts you at the 1 mile circumference point on the earth.

i'm no geometry sphere expert, so someone will have to let me know if that is physically possible (i.e. i tend to think that if you walk n units south from any point on the northern part of a sphere, other than the north pole, it is impossible for the circumference to be n or less than n, but who knows?)

finally there are actually an infinite number of points. if we consider the case before where we went to the point with a circumference of 1, why not go to the point with a circumference of 1/2. then when you go a mile east, you loop around twice, and end up in the same spot. this holds true for 1/3, 1/4, 1/5, ... 1/n, etc.