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?

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?

• Hint: one on the northern hemisphere and infinitely many in southern hemisphere. – achille hui Oct 30 '16 at 4:25

As @achillehui's comment pointed out, there are an infinite number of such points.

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.

• @achillehui Thank you. – Deepak Oct 30 '16 at 4:49
• @Deepak Is there no such latitude circle(of circumference of one mile) in northern hemisphere? – Pulkit Jun 13 '17 at 5:44
• @Pulkit: No. After walking South for one mile your distance to the North Pole is at least 1 mile. Therefore the circumference of the latitude you are on is at least close to $2\pi$ miles, and $2\pi >1$. – Jyrki Lahtonen Jun 13 '17 at 8:11