Famous Problem

Geometry Level 2

A man walks 1 mile south, 1 mile east, and 1 mile north. He ends up exactly where he started. At how many points could he start?

Assume the earth is a solid sphere.

\infty

1 0

3 solutions

Tunk-Fey Ariawan
Mar 1, 2014

We all know that south is perpendicular to east and east is perpendicular to north. If we start to walk to south, then to east, and then to north with equal distance in Non-Euclidean geometry , the track will form a equilateral triangle with all three of its internal angles are $90^\circ$ , summing to $270^\circ$ . Take a look the picture below.

270 Degrees Triangle

In conclusion, there are actually an infinite number of points such that we will end up exactly at the same spot where we start regardless where we start to walk. $\text{\# }\mathbb{Q.E.D.}\text{ \#}$

To those saying that infinity is not the correct answer, read my solution. Unfortunately this solution itself is wrong (the angle made between the walk south and the walk north is not necessarily $90^\circ$ ; just see if you walk somewhere near the equator, you'll find it closer to $0^\circ$ ; besides, the angle has no bearing to the solution), so I guess wrong reasoning leading to a correct answer here.

Ivan Koswara - 7 years, 2 months ago

"In conclusion, there are actually an infinite number of points such that we will end up exactly at the same spot where we start regardless where we start to walk."

Well, start anywhere near equator, we will clearly end up more east from start point, therefore your solution is incorrect.

It's not possible for us to end up exactly same point starting somewhere between north pole and approximately $1+\frac{1}{2\pi}$ mile from south pole. Read Ivan Koswara's solution below here for more details.

Micah Wood - 7 years, 2 months ago

The answer is NOT RIGHT. There's only one place - north pole. If you even start an inch away from the pole, then you will end up being a tiny bit away from the pole at the end, because you won't actually make a complete triangle.

Fryderyk Wysocki - 7 years, 3 months ago

Ok, for future reference: Tunk-Fey's explanation is wrong. That said, my explanation doesn't take the infinite number of points that Ivan talks about in his explanation.

Fryderyk Wysocki - 7 years, 2 months ago

Ivan Koswara
Mar 22, 2014

Take a point about $1 + \frac{1}{2\pi}$ miles from the south pole. (I'm not sure about the exact number, but bear with me.) A walk south means you're now about $\frac{1}{2\pi}$ miles from the south pole. Here's the trick: walking 1 mile to the east means you circle the south pole exactly one circle, ending up back where you were before walking east! And finally you can walk up $1$ mile, arriving back where you started.

All those points that are that distance from the south pole work, and there are certainly an infinite number of points satisfying that (all points in a circle). Thus the answer is infinite .

Also, observe that you can circle any positive integer number of times around the south pole, giving various other starting points. Lastly, of course, the popular north pole answer, where here you stick $1$ mile from the north pole.

I believe this is the only correct solution.

Yong See Foo - 7 years, 2 months ago

Krishna Garg
Mar 2, 2014

Since he completes his journey from where he started,there could be any number of points all over the earth being sphere,therefore answer is infinite.

K.K.GARG,INDIA

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