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# Challenge Question: Global Adventurer (March 2022)

Posted February 28, 2022 12:00 AM
Pathfinder Tags: challenge question

Describe the locations on Earth that would make the following statement true?

"I traveled 1 mile north, 1 mile west, 1 mile south, and ended up exactly where I started."

We'd appreciate an explanation (or would accept an equation).

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#1

### Re: Challenge Question: Global Adventurer (March 2022)

02/28/2022 3:45 AM

Start at the South pole: no explanation needed.

Or start any where on a circle centred on the North pole with a radius of:

1 + 1/2nπ

When n is 1, the first mile brings you to the circle with circumference 1 mile around the pole; the second mile just takes you round that circle, and, the third takes you back to your starting point.

When n is 2, you just have to walk around that inner circle twice etc.

Of course as n approaches infinity you just start 1 mile from the North pole, walk to it, get very dizzy, then, return to your starting point.

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#4

### Re: Challenge Question: Global Adventurer (March 2022)

03/03/2022 11:02 AM

This is stretching it, but there is another possibility. If, when you begin your 1 mile walk west you are on the edge of a narrow gorge (or a tall skinny mountain) that is one half mile deep, your 1 mile walk west will be 1 mile that is almost nothing in the x,y plane. In other words, your 1 mile walk will be mostly vertical and your horizontal travel will be close to zero. Then your return south will bring you back to where you started..or close.

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#31

### Re: Challenge Question: Global Adventurer (March 2022)

03/28/2022 11:24 AM

You are certainly the closest and quickest to it.

The answer is (∞ + ∞ + 1). Or just plain ol' infinity.

1. The South Pole is the easiest to decipher. Your walking path looks like a triangle.

2. Any point on Earth that is 1 + ½Π miles from the North Pole is also correct. This permits you to complete one circumference that is 1 mile long to return to the end of the northern path, and return to your starting point. Your exact starting position is about 1.159 miles away from the pole.

3. You could also move increasingly closer to pole, and do multiple circumferences around the Earth. So if you wanted to do two laps, instead of one, your starting point would be 1 + ¼Π miles from the North Pole. This is theoretically infinite as well, as you can just keeping getting closer to the pile. Practically speaking, you'd be really dizzy.

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#2

### Re: Challenge Question: Global Adventurer (March 2022)

03/03/2022 4:25 AM

At the South Pole.....but it could be anywhere else on Earth.... pending a compass re-calibration......

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#23

### Re: Challenge Question: Global Adventurer (March 2022)

03/10/2022 8:10 AM

There are other possibilities, though.

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#3

### Re: Challenge Question: Global Adventurer (March 2022)

03/03/2022 4:52 AM

Can only be the South Magnetic pole because the word North is used.
If the question was "I traveled 1 mile south, 1 mile west, 1 mile north, and ended up exactly where I started." .......it can only be the North Magnetic pole.
Bloody compass's ..... other forms of navigation instrument COULD make it simply the Pole.

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#5

### Re: Challenge Question: Global Adventurer (March 2022)

03/03/2022 11:33 AM

South pole and only the south pole.

Your first leg of you journey will follow the "Meridians" or "lines of longitude"

Your third leg will bring you back to where you started by following the lines of longitude.

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#6

### Re: Challenge Question: Global Adventurer (March 2022)

03/03/2022 11:46 AM

Sorry to disagree, but I think Randall is correct. You start at a point that is south of the north pole so that a one mile walk north takes you to a latitude whose circumference around the earth is one mile. You walk that mile west and you end up exactly where you started where you began that one mile walk west. Then you walk south 1 mile, essentially back tracking your initial one mile walk north.

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#7

### Re: Challenge Question: Global Adventurer (March 2022)

03/03/2022 1:37 PM

That's fine to disagree, my point is, if you were to start anywhere more northernly other than the south pole, your lines of longitude will not intersect your starting point.

I'll be listening...

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#8

### Re: Challenge Question: Global Adventurer (March 2022)

03/03/2022 2:02 PM

Sorry, I was pulled away to be able to finish...

if you started anywhere other then the south pole, on your third leg of your journey the path would be to the west of your starting point.

Because your lines of longitude will not intersect your starting point but will go directly to the south pole. (west of your starting point)

and the path would be to the west of your starting point....

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#9

### Re: Challenge Question: Global Adventurer (March 2022)

03/03/2022 2:20 PM

Explain to me if you walk one mile west on a latitude whose circumference around the earth is one mile, why you would not end up at exactly the same longitude you started out at?

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#11

### Re: Challenge Question: Global Adventurer (March 2022)

03/03/2022 2:48 PM

Because the longitude lines only converge at two (2) places... one is at the north pole, and the other is at the south pole. (not to size... lol) and I only have north, east, west and south arrows... but I think you can get the picture...

again...

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#10

### Re: Challenge Question: Global Adventurer (March 2022)

03/03/2022 2:36 PM

Maybe this helps?

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#12

### Re: Challenge Question: Global Adventurer (March 2022)

03/03/2022 2:56 PM

after your first leg of your journey, when you arrive directly on the north pole... literally... from there all directions from the north pole goes to the south pole... until after you take your first step...

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#13

### Re: Challenge Question: Global Adventurer (March 2022)

03/04/2022 5:00 AM

You never arrive at the North pole!

You start anywhere 1 + 1/2π away from the North pole; after the first leg of your journey you are 1/2π away from the North pole; after the second leg in which you walk all the way around the pole, you are back at the same point, and your third leg takes you back to your starting point.

For the pedantic the minuscule correction you would need to make for the curvature of the earth would be tiny compared to the size of the human foot, and so, is irrelevant. But you would need to take it into consideration in Northern hemisphere triple journeys of say five thousand miles.

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#14

### Re: Challenge Question: Global Adventurer (March 2022)

03/04/2022 7:47 AM

So here's a slightly more difficult problem.

If you do the triple journey from the South pole, the third leg is completely different from the 1st leg. Whereas if you do one of the triple journeys at the North pole the third leg is always exactly the reverse of the first leg.

There is a triple journey (x miles N, x miles W, and x miles S) which starts and ends at the South pole and in which the 3rd leg is the reverse of the 1st leg. Find x?

I cheated and used a spread sheet to get an answer, but, surely there's a way to solve this without a spread sheet or other successive approximation technique.

I'll post my crappy solution after giving others a while to think about it.

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#15

### Re: Challenge Question: Global Adventurer (March 2022)

03/05/2022 11:27 AM

There is a triple journey (x miles N, x miles W, and x miles S) which starts and ends at the South pole and in which the 3rd leg is the reverse of the 1st leg. Find x?

Well, the distance you walk North is x radians, the circumference of the earth walking along a constant latitude is 2*pi*sin(x) radians. Where the two curves cross is the point where all three legs would be equal.

Plotting y=x and y=2*pi*sin(x):

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#16

### Re: Challenge Question: Global Adventurer (March 2022)

03/05/2022 11:31 AM

Huh?
We, me and my engineers have no idea what you have been smoking/drinking....

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#17

### Re: Challenge Question: Global Adventurer (March 2022)

03/05/2022 5:15 PM

A picture might help.

For simplicity, let the earth radius = 1. We can scale it back up later.

Walk north from the south pole a distance x. Your distance from the axis of the earth would be sin(x), and if you turned west and walked around your line of latitude, the distance would be 2*Π*sin(x). We want the value of x such that all three segments are equal, i.e. x = 2*Π*sin(x).

The blue line is the distance walked north (and south). The red curve is the distance walked west at constant latitude as a function of the distance walked north. At the intersection the distances north, west, and south are equal.

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#18

### Re: Challenge Question: Global Adventurer (March 2022)

03/06/2022 6:06 AM

That's what I got:-

We both got 2.6978 radians, so the slight difference will be because of a difference in the value of the radius of the earth.

But we both had to use graphs/successive approximation/inspection.

Surely there's a way to solve

2πsin(θ)/θ = 1 analytically

I'm sure I've seen discussion of the function Sin(θ)/θ before. But all I can find now is proof that it is 1 at θ=0.

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#21

### Re: Challenge Question: Global Adventurer (March 2022)

03/10/2022 8:03 AM

The problem is that the direction <...west...> is not defined for latitudes 90oN and 90oS. This might be throwing those thinking about only the South Pole as the answer.

However, if the walk <...1 mile west...> were a positive integer number of turns at that latitude that ended up at the same longitude on completion, then that concept is correct.

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#24

### Re: Challenge Question: Global Adventurer (March 2022)

03/10/2022 8:11 AM

There are other possibilities, though.

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#19

### Re: Challenge Question: Global Adventurer (March 2022)

03/08/2022 7:47 AM

In the statement - " 1 mile north, 1 mile west, 1 mile south", that means you made two 90 degree turns. So you made a triangle with three 90 degrees angles. The last part - "ended up exactly where i started" confirmed that this is possible only on a globe Earth.

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#20

### Re: Challenge Question: Global Adventurer (March 2022)

03/08/2022 7:52 AM

Unless of course you used a GPS instead of using a simple compass…

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#22

### Re: Challenge Question: Global Adventurer (March 2022)

03/10/2022 8:10 AM

Might one assume a smooth surface to walk on for this challenge?

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#25

### Re: Challenge Question: Global Adventurer (March 2022)

03/10/2022 8:20 AM

There is only one place in the Southern Hemisphere where this would work. There are an infinite number of places in the Northern Hemisphere.

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#26

### Re: Challenge Question: Global Adventurer (March 2022)

03/12/2022 5:04 PM

There's only one starting spot and ending spot in the Southern Hemisphere, but a lot of northerly directions you could start walking.

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#27

### Re: Challenge Question: Global Adventurer (March 2022)

03/14/2022 10:35 AM

on the same direction with this thread about North, this has been reported about the past couple of years, but on a more recent report. that the magnetic North is moving towards Russia.

wasn’t it also reported that the poles are about to invert to each other?…

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#28

### Re: Challenge Question: Global Adventurer (March 2022)

03/14/2022 12:14 PM

The poles do periodically flip, and I guess that we could be subject to some nasty effects from the sun when they do.

But also for a few years the Russians believed that there was a real possibility that the whole earth might flip due to the Dzhanibekov effect.

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#29

### Re: Challenge Question: Global Adventurer (March 2022)

03/14/2022 1:06 PM

Yes, the poles do periodically flip, how they found out is the ocean floor they determined that the magnetic polarity have flipped on the the continued creation of new crusts on the ocean floor

I never heard the Dzanibekov effect being applied to the earth actually flipping...

Pretty interesting...

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#30

### Re: Challenge Question: Global Adventurer (March 2022)

03/15/2022 1:56 PM

If you determine you directions by GPS (hope the GPS system is working down under) start at the South pole.

Choose any direction and continue on that same course.

Then turn west and move west . You will follow a curved path as seen from top.

Then turn South and you will reach your starting point .

Trying to determine directions by compass would not work well.

PS use thick/warm socks.

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