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        riddles >> medium >> Globe Traversal
        
(Message started by: Charon on Aug 6^th, 2002, 11:50pm)

Title: Globe Traversal
Post by Charon on Aug 6^th, 2002, 11:50pm

I have been able to come up with one solution that would really allow an infinite amount of positions allong specific latitude's

Remember
Radius: r
Diameter: d
Circumference: C

C = 2 Pi r = Pi d

So if C= 1(miles) then r = 0.15915.......(miles)

If you were to stand 1 + r(miles) from the south pole then walk south 1 mile you would be r(miles) from the pole and if you were to travel one mile west you would completely circle the south pole once and end up at the same position. You can now travel one mile north to your starting point.
You can allso divide r by 2 giving you r= 0.079577.....
and you would have to circle the south pole twice while traveling west. You can continue to find values for r that will allow you to circle the south pole and end up in the same position to travel north from. Giving you plenty of points to start on specific latitude's that will allow you to finish where you began.

Are there any other solutions or a flaw in my own ?

Title: Re: Globe Traversal
Post by Guest on Aug 8^th, 2002, 1:49pm

Add the obvious choice: Start at the North Pole

1 mile south + any amount west + 1 mile north, and you are back where you started

Title: Re: Globe Traversal
Post by Mungbeam on Sep 2^nd, 2002, 4:18pm

Hmm... my only complaint is that it's not obvious that walking a mile west on the south pole means you end up in the same place.

What I mean is, if I'm at the south pole, I check my compass, find west, take a step. West now changes, so I adjust direction, and take another step. I'm going round in circles, but it's not obvious that I'll face the same direction I came from... and all directions are north at the south pole, so I could quite easily walk south 1 mile, do a little circuit around the pole, and start north up the opposite side of the globe.

Then again, it's still POSSIBLE to end up at the same place, so there's the north pole, and every point one mile north of the south pole, if you allow for that fact that someone not trying to get back to the same point probably wont.

Title: Re: Globe Traversal
Post by DeeK on Sep 9^th, 2002, 7:33pm

on 09/02/02 at 16:18:10, Mungbeam wrote:

Hmm... my only complaint is that it's not obvious that walking a mile west on the south pole means you end up in the same place.

Hi Mungbeam, I don't think you understood the answer properly. The solution in the first post doesn't have anything to do with actually walking _on_ the south pole. You walk in a circle _around_ the south pole.

Imagine that there is a circle exactly 1 mile in length, with the south pole as the center. If you walked around that circle for a mile, you'd end up in exactly the same spot. And THAT is the trick to the question.

Title: Re: Globe Traversal
Post by Kozo Morimoto on Sep 10^th, 2002, 1:58am

Or any of the smaller circles within that original circle where by the circumference is 1/2 (meaning you do 2 laps around the pole) or 1/3 (meaning you do 3 laps around the pole) etc etc

Title: Re: Globe Traversal
Post by Mungbeam on Sep 15^th, 2002, 2:21am

*slap forehead*

I see now... thanks!

Title: Re: Globe Traversal
Post by sks2141 on Jan 23^rd, 2007, 5:21am

I am thinking on the following lines based on sphere property and vectors:

Case 1) If we start exactly on the north pole:

if one moves south 1 mile, => one moves along the part of the circumference of the earth

n then west or east one mile => one moves along the lines which is perpendicular to the axis of earth (under consideration) and so, this wont make any difference whether one moves 1 mile or greater
coz the cross product will be zero

n then north one mile => one will reach the north pole .... because we have started from the exact north pole and considering the basic property of sphere

Case 2) If we started on any other point except the extreme poles .... say equator.

if we move south 1 mile, then east or west 1 mile n finally north 1 mile , then depending upon the surface parameters , we will never end up in the same place ..... it will be a kind of a quadrangle on the spherical surface considering joining the start point and the end point

Case 3) If we started on the south pole

Considering Space as Reference Only:
if we move south 1 mile => we are actually moving north one mile considering seeing from the space as reference and not the moving object !!!
so now, if one moves 1 mile east or west and then north one mile :=> one is moving even further north and will not come on the same place

Title: Re: Globe Traversal
Post by Icarus on Jan 23^rd, 2007, 4:14pm

Except that you are wrong about case 2. While it holds near the equator, there are infinitely many locations where it is false. Read the earlier posts (or the other threads for this riddle) to find out where.

And case 3 is simply impossible (more generally, it is impossible anywhere within 1 mile of the South pole). You cannot travel south from the south pole, so as soon as you are there, you are no longer able to act as the puzzle says.