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Walking 3 kilometres on a planet like Earth. Question
Imagine you are on a planet the same size as Earth but this planet is perfectly spherical and solid everywhere. Furthermore assume it has a North and a South Pole just like Earth and has directions of north, south, east and west just like Earth.
You start somewhere on this planet and walk one kilometre south, then one kilometre east and lastly one kilometre north.
Is it possible that you end up where you started from?
If the answer is yes, describe all possible starting positions.
If the answer is no, explain why it is impossible.
4 answers
The following users marked this post as Works for me:
| User | Comment | Date |
|---|---|---|
| will.octagon.gibson | (no comment) | Mar 11, 2026 at 19:16 |
Yes, an infinite number of ways.
Description
Lundin and Monica have already described starting at the north pole, and Monica added a second method near the south pole.The south pole method can be expanded to an infinite number of rings that have a circumference of 1 km divided by any positive integer. You go south 1 km to hit one of these rings. Then you go around the ring a whole number of times to end up where you started on the ring. Then 1 km north gets you back to your original starting point.
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The following users marked this post as Works for me:
| User | Comment | Date |
|---|---|---|
| will.octagon.gibson | (no comment) | Mar 11, 2026 at 19:16 |
Yes, multiple ways.
Explanation
You can start at the north pole. Walking 1km north from any point in the ring at 1km south of the pole leads back to the pole, so it doesn't matter how far east or west you walk.
At the south pole, you want to find the latitude ring that is 1km in circumference, slightly north of the pole. Start your walk anywhere that is 1km north of that ring. You'll walk south, hit the ring, go east and do a complete loop, and then walk north to your starting point.
Further, you can do this with the latitude ring that is 0.5km in circumference; you'll walk twice around and then return to your starting point. You can walk three times around at 0.333...km, four times at 0.25km, and so on. Any starting point that allows you to walk 1km around a ring and return to the exact same place on the ring (before returning north) works. Introductory calculus was a long long time ago, so I'm afraid I can't express this mathematically.
1 comment thread
Yes it is possible.
Reasoning.
Obviously this will happen if you start at the (geographic) north pole (and only there). If you first walk n kilometers south, then an arbitrary distance either east or west, and then n kilometers north, you end up on the north pole again.
The same would apply to the south pole except you can't walk 1km south starting from the south pole by definition.
If you start anywhere else you'd merely be walking along the longitude/latitudes in an 'U' shape.
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