Walking around the world?

Assume for this problem that the earth is a perfect sphere. A point $P$ on it's surface has the following property. Starting from $P$, suppose you travel $1$ mile south, then $1$ mile west and finally $1$ mile north. Doing so takes you back to the same point $P$. Are there any such points $P$ other than the north pole? Choose the appropriate option:

• There are no such points.

• Such points for a single circle.

• Such points form at least two but finitely many circles.

• Such points form infinitely many distinct circles. Note by A Brilliant Member
5 years, 8 months ago

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"Such points can form infinitely many distinct circles", because one can start at a point that is 1 + x miles from the South pole, travel 1 mile south to a distance x from the South Pole, such that in travelling west 1 mile one can end up making n number of circuits coming back to exactly where one was before travelling west, and thence travel 1 mile north back to the point of beginning P. n can be any integer.

- 5 years, 8 months ago

Can you clarify what you mean by "making n number of circuits"?

- 5 years, 8 months ago

x = 1 / 2πn, where n is any integer > 0. Then n circuits or trips around will bring him back to the point where he began traveling west.

- 5 years, 8 months ago

There are no such points.

- 5 years, 8 months ago

No. Check your reasoning.

- 5 years, 8 months ago