rescue operation - where is your partner?

Your partner's space ship has crashed on an uninhabited planet. Only the radio transmitter and his compass were still in operation. He asks you to rescue him, and tells you how to find him by the following story.

1. He went one unit to the south
2. Then he went one unit to the west
3. Then he went one unit to the north
4. After that journey, he was back at the beginning where he started

Where is your friend? Can you really find him?

• Well, let's say it's remarkable to speak of north and south of unknown planets - not even all of them have a magnetic field that would let you deduce which way is north to begin with. – Tim Couwelier Oct 7 '14 at 13:10
• @TimCouwelier The shocking twist is that it is earth, and all the humans are dead! – Golden Dragon Oct 7 '14 at 13:11
• Note that the usual formulation of this riddle asks "What color is the bear" demonstrations.wolfram.com/WhatColorIsTheBear – yoniLavi Oct 7 '14 at 15:15
• And to complete @yoniLavi's comment, the correct answer is "black". It's a trick question which catches out everyone who believes that polar bears are white. – Peter Taylor Oct 7 '14 at 15:47
• And to the question: "where is your friend?" The answer is obviously "At the crash site". – Florian F Oct 8 '14 at 20:01

I believe the complete answer is the following:

• He could be at the north pole
• He could be near the south pole, exactly one unit north of a place where the parallel is one unit long.
• He could be one unit north of a place where the parallel is $1/2$ unit long.
• and so on wih $1/3$, $1/4$,...

The point being when he goes west, he could be that he's making a total round (or several), before going north back to his original place.

• He could still be in the middle of nowhere... – Cullub Oct 7 '14 at 12:17
• you knew this - thanks for propsing that solution (even it's kind of sad no more pictures have been added)!! very good! – Martin Frank Oct 7 '14 at 12:18
• actually, the physics geek in my feels like pointing out that it's even more complicated then this. If he is that close to a pole then how he measures 'north' and 'south' is complicated. really no measurement is trustworthy, but ignoring that imagine he was looking at his compass as he walked a unit. The compass would constantly adjust the direction 'north' was as he moved, since this close to a pole a unit travel effects direction, causing him to walk in a slight arc. This suggests potential for a more complicated puzzle involving arcing movement and geometry..I would love to see one ;) – dsollen Oct 7 '14 at 19:14
• @dsollen yes, right, maybe i must specify that this planet is very special and has a very 'precise' magnetic pole! it's not like earth, i think aliens were involved, and it's totally working with that compass... – Martin Frank Oct 8 '14 at 10:18
• So...an Alien did it? (tvtropes.org/pmwiki/pmwiki.php/Main/AWizardDidIt) :) – dsollen Oct 8 '14 at 15:15

Where he was: In the middle of nowhere.

He started there, went three directions, and was still there when he stopped.

• very nice - i really like that!! – Martin Frank Oct 7 '14 at 12:16

If the planet is only 1 unit around, but more than one tall, he could be anywhere on the planet.

• sorry i forgot to mention, it's a totally spherical planet, its size is ... huge ... – Martin Frank Oct 8 '14 at 4:12
• @MartinFrank So? Maybe the unit is huge. This answer is perfectly consistent with the question. – Gilles 'SO- stop being evil' Oct 13 '14 at 10:49
• very clever @Gilles but that's not 100% right - when taling of a huge planet it says "the planet has a huge amount of units" (common sense of a huge planet) BUT you can argue with me that it's not been pointed properly out before... – Martin Frank Oct 13 '14 at 11:08
• @MartinFrank what he's saying is that the question only says the planet is "unihabtiated" (sic), you then said "it's spherical and ... huge" which still isn't specific. "Huge" could be "1 unit around", if it's a big unit. For example, 1 AU circumference is a huge planet, but 1cm is a tiny planet. The unit was not specified, so 1 unit could well be whatever "1 circumference" happens to be – Joe Oct 13 '14 at 12:40
• With all that said, if it's spherical then my answer doesn't work anyway. I'll still leave it here because that wasn't clarified at the time of answering :-) – Joe Oct 13 '14 at 12:41

He could be at the North Pole:

• yes, very right - he could be at the north pole!! – Martin Frank Oct 7 '14 at 11:43
• *gg i didnt know the north pole is in saudi arabia ^^ – Martin Frank Oct 7 '14 at 11:44
• @MartinFrank ... whoops, I probably should have picked a better image. :D – Doorknob Oct 7 '14 at 11:44
• it's really ok, it helps to point out your idea - the idea is very good (although i heard there might be another location?!?) – Martin Frank Oct 7 '14 at 11:45

Think about a circle that is 1 unit in circumference, and just north of the South Pole. We’ll call that C(1).

If you are 1 unit north of this circle, then you will also end up back at the same spot. You will travel one unit south, then you’ll travel one unit around the circle, and then you’ll go north and end up back at the same spot as you started.

Actually this is true for any point on the circle that is 1 unit north of C(1). So this is an infinite number of solutions.

Infinity times infinity!

Now think about a circle that is 1/2 unit in circumference and also just north of the South Pole. We’ll call that C(1/2).

If you are 1 unit north of this circle, then you will also end up back at the same spot. You will travel one unit south, then you’ll travel the circle TWICE, and then you’ll go north and end up back at the same spot.

Similarly, if you are one unit north of the circle C(1/3)–a circumference of 1/3 near the South Pole–then you will also end up back at the same spot. You will travel around C(1/3) a total of 3 times.

We can use the same argument for C(1/4), C(1/5), and so on for any C(1/n), where each circle has a circumference of 1/n. If you go one unit west in any of these circles, you will travel around the circle n times and end up back at the same point.

Therefore, you will always end up at the same spot if you are one unit north of any circle C(1), C(1/2), C(1/3), etc.

There are an infinite number of C(1/n) circles, and you can be anywhere on the circle one mile north of each circle.

So this is basically infinity times infinity points.

In conclusion

The complete solution is you can be: one unit north of C(1) (infinity points) one unit north of C(1/2) (infinity points) one unit north of C(1/3) (infinity points) … one unit north of C(1/n) (infinity points) …

So this is basically 1 + (infinity)(infinity).

So no, according to this conclusion you wont be able to find him.

• Why the down votes? – c.pullen Aug 25 '16 at 8:11

It's a trick question!

If your partner can tell that they're in precisely the same location after all of these movements, then they must have a precise measure of their initial location. A unit could be thousands of miles, or 1 step, but to know they're in the exact same locaiton, they need to have a solid indication of where they are. If they have that information then they're wasting time telling you about moving by an undefined unit and probably don't want to be found.

If they're wasting your time and theirs talking about moving about when they have a precise co-ordinate to begin with, I wouldn't bother.

P.S. I know it's the north pole, but they don't know that from the compass. Also, if you were right at magnetic north then you would probably notice the compass working very erratically.

• It's fairly easy to see that you are standing next to your broken spaceship. As such the claim of having to know where he is just doesn't make sense. – Dennis Jaheruddin Oct 7 '14 at 14:01
• @AJFaraday: This is wrong for so many reasons. Nothing bugs me more than someone trying to give a tricky answer that isn't even right. (1) Yes, you can tell if you are next to your spaceship (2) There's nothing inconsistent about being able to precisely measure displacements and directions, but not location (3) It's not necessarily only the North Pole (4) If you were at the North pole, your compass would not really behave erratically, it would want to point up and down.... – ThePopMachine Oct 7 '14 at 19:35
• ... If you could magically hold it perfectly perpendicular to the magnetic field, it wouldn't move. If it was slightly inclined and the force (torque technically) is enough to overcome the friction of the needle against the axle, it would point in the direction of the incline. – ThePopMachine Oct 7 '14 at 19:37
• @ThePopMachine Well, I'm very sorry to have annoyed you. – AJFaraday Oct 8 '14 at 9:08
• @AJFaraday: Sorry I went off. But I believe this SE doesn't deserve to come out of beta and should be closed because it is just flooded with everyone wanting to give their tricky answer and prove how clever they are. That or the mods need to create some guideline and enforce it. Like they always reply with boiler plate "Welcome to puzzling.SE: This is not the same as other forum sites. Here we strive to find the best answers to questions. Please don't post your "trick answers" to question which aren't looking for tricks. " – ThePopMachine Oct 8 '14 at 14:44

Apart from the obvious north and south pole options, i went for some more trivial answers:

On the equator

One unit south, any amount of units east and then one unit north will always bring you back to (a different point along) the equator.

Alternately, if that unit happens to be the exact circumference of the earth, one unit south from the equator will be a full circumnavigation, followed by another circumnavigation around the equation and a third circumnavigation, ending up at the same point.

multiple north poles

The journey starts at the magnetic north pole and finishes at the geographical north pole... This could (technically)(not really) count as the same place on a map...