Mars rover

[Genady]

Mars rover made 100 km South, then 100 km West, and then 100 km North, arriving to the starting point. What was the starting point?

This puzzle is easy, but not too easy.

[iNow]

12 hours ago, Genady said:

Mars rover made 100 km South, then 100 km West, and then 100 km North, arriving to the starting point. What was the starting point?

This puzzle is easy, but not too easy.

Spoiler

100 km West ? 

 

[Genady]

1 hour ago, iNow said:

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100 km West ? 

 

Spoiler

I am sorry, but I don't understand, which point you meant.

 

[TheVat]

18 hours ago, Genady said:

Mars rover made 100 km South, then 100 km West, and then 100 km North, arriving to the starting point. What was the starting point?

 

Spoiler

You don't say actual linear distance Rover travels, just a vector.  So journey begins and ends at one acute angle of right triangle. The last part of trip is along hypotenuse, covering 141.42 km.

Finally figured out how to get hide button to work on tablet.

Edited May 12, 2023 by TheVat
getting hide button to work arghhhhhhh

[Genady]

19 minutes ago, TheVat said:

Dammit, I can't find the hide button on an Android tablet.  So I will just say (LIKELY SPOILER)

 

 

 

 

 

 

You don't say 100 km is the actual distance the rover traverses on each part of its journey.  That could be the magnitude of one vector, on each part of trip.  If so, easy peasy lemon squeezy.

 

100 km on its odometer, three times. Each leg's length is 100 km. 

Edited May 12, 2023 by Genady

[TheVat]

Ah, thanks.  Now I see the actual problem.

Spoiler

So starting point is north pole.  

 

 

[Genady]

1 minute ago, TheVat said:

Ah, thanks.  Now I see the actual problem.

  Reveal hidden contents

So starting point is north pole.  

 

 

Spoiler

Good. But. That was the easy part.

Now, find another point.

 

[TheVat]

Spoiler

Oh, you mean the other points on the trip from the pole?  Well, they would be along whatever line of latitude was 100 km from the pole.  Or you are asking if there are other potential starting points?  

 

[Genady]

8 minutes ago, TheVat said:

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Oh, you mean the other points on the trip from the pole?  Well, they would be along whatever line of latitude was 100 km from the pole.  Or you are asking if there are other potential starting points?  

 

Spoiler

Yes, other potential starting points. Other than the North Pole. Hard to believe, I know ...

 

[TheVat]

Spoiler

Hard to see that, with a fixed coordinate system.  The spherical geometry seems problematic, because latitude lines aren't great circles, except for the equator.  Hmm.

 

[Genady]

26 minutes ago, TheVat said:

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Hard to see that, with a fixed coordinate system.  The spherical geometry seems problematic, because latitude lines aren't great circles, except for the equator.  Hmm.

 

Spoiler

You're right. However, the definitions are unambiguous. Directions North and South are great circles from pole to pole. Directions West and East are circles parallel to equator. Hint: it is not really difficult, not much geometry involved, just thinking ...  

 

[Ken Fabian]

Spoiler

I got the obvious one. Less obvious - took me a while - there is a line of latitude around the South Pole with a circumference of 100km. Starting point is any point 100km North of that line - a 100km leg South, then 100km West in a circle to return to the same place, then North.

 

[Genady]

10 minutes ago, Ken Fabian said:

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I got the obvious one. Less obvious - took me a while - there is a line of latitude around the South Pole with a circumference of 100km. Starting point is any point 100km North of that line - a 100km leg South, then 100km West in a circle to return to the same place, then North.

 

Spoiler

Right. Right. The North Pole and all points about 116 km North of the South Pole.

Now, other point / points.

 

[TheVat]

2 hours ago, Genady said:

You're right. However, the definitions are unambiguous.

Ahhh. 

Spoiler

Then I think is needed a line of latitude a short distance from the South Pole such that its circumference would be 100 km.  Travel south to that line 100 km, then go west full circle, then go north again.  I didn't think about retracing steps but I see that's allowed.  Ha!  Start at any point 115.92 km north of south pole.

Another fine puzzle!  I need to unrust my math brain, so these are delightful.

 

[Genady]

9 minutes ago, TheVat said:

Ahhh. 

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Then I think is needed a line of latitude a short distance from the South Pole such that its circumference would be 100 km.  Travel south to that line 100 km, then go west full circle, then go north again.  I didn't think about retracing steps but I see that's allowed.  Ha!  Start at any point 115.92 km north of south pole.

Another fine puzzle!  I need to unrust my math brain, so these are delightful.

 

Spoiler

Very good. You got the North Pole and all the points 115.92 north of the South Pole.

However, nobody got +1 yet, because ... there is / are still other possible point / points!

 

[npts2020]

7 hours ago, Genady said:

Very good. You got the North Pole and all the points 115.92 north of the South Pole.

However, nobody got +1 yet, because ... there is / are still other possible point / points!

Spoiler

1. you have the circle 100 km north of whatever latitude where the planet is 100 km in circumference but also any latitude north of where the circumference divides into 100 evenly.

 

Edited May 13, 2023 by npts2020
clarification

[Genady]

56 minutes ago, npts2020 said:

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1. you have the circle 100 km north of whatever latitude where the planet is 100 km in circumference but also any latitude north of where the circumference divides into 100 evenly.

 

Spoiler

Right. I don't know of any other points. +1

 

[TheVat]

Spoiler

Actually, Rover could depart from 100 km north of the south pole, arrive at pole, then make tight counterclockwise circles there until the odometer shows another 100 km, then return to start.  The mission runners at JPL would probably assume their Rover had gone nuts.   😀

Plus one from me, too, to Npts for seeing the next step.  

 

[Ken Fabian]

@npts2020, @Genady I gave up and peeked. But I have to admit not seeing how that third solution can work; seems to me that two different N/S lines from other circles of latitude will converge at the pole and won't converge at 100km.  Can one of you show it diagram style?

Edited May 14, 2023 by Ken Fabian

[Genady]

1 hour ago, Ken Fabian said:

@npts2020, @Genady I gave up and peeked. But I have to admit not seeing how that third solution can work; seems to me that two different N/S lines from other circles of latitude will converge at the pole and won't converge at 100km.  Can one of you show it diagram style?

Spoiler

Starting 108 km North of the South Pole (point A), makes 100 km South (point B). The circumference around the South Pole there is 50 km. The rover makes 100 km West = two full circles, back to the point B. Then 100 km North back to the point A.

Similarly, starting about 105 km North of the South Pole, South to the circle of circumference 33.3 km, three full circles around the South Pole, and back to the starting point.

Etc. OK?

 

[Ken Fabian]

@Genady Ah, yes, thanks.

Spoiler

For some reason I got stuck on thinking multiples of 100km, not fractions.