PDC and a Nice Geographic Puzzle

Hi Folks,

At PDC I gave a talk largely inspired by topics raised here and in the spatial forums.  But “inspired by” doesn’t equate to “a duplicate of”, and to turn things around, I’ve been meaning to write a few posts here inspired by my PDC talk.  Stay tuned.

In the meantime, Rob Mount from Intergraph sent me a note that started:

As I review your PDC presentation I’m reminded of a puzzle I think you’ll enjoy. 

I did indeed, and I hope a lot of you will enjoy it as well.  Rob started out with a puzzle I expect most of us have heard before:

When I was growing up one of my uncles was fond of challenging the children in the family with puzzles.  One of my favorites, as a very young child, was this one: A hunter walks a mile south, a mile east and a mile north and finds himself back at the starting point.  A bear walks by.  What color is the bear?

This is well-known enough that I don’t think the answer will be much of a spoiler:

He, of course, thought the hunter started at the North Pole.  The bear was a polar bear and hence white.

A perfectly valid answer, of course.  But Rob continues:

Years later I finally got my revenge by stumping him with this problem:  I reminded him of the hunter and the polar bear and pointed out that there are actually several other points on the earth that meet the geographic constraint he stated – if you walk a mile south, a mile east and a mile north you find yourself back where you started.

Assuming a spherical earth, how many such points exist and where are they?

I won’t spoil this one so quickly.  Go ahead and post your answers in the comments; I’ll post the solution in a few days.

Cheers,
-Isaac

Comments

• Anonymous
  December 05, 2008
  4, right?  One for each pole and one each for the magnetic poles. And, of course, there's no polar bears in Antarctica :-)

• Anonymous
  December 05, 2008
  The comment has been removed

• Anonymous
  December 05, 2008
  I can only think of one at the moment.  Start a bit more than a mile north of the south pole, such that when you go south one mile, you're at a point where going east one mile will get you back where you were (by walking around the south pole), then you can go north again and get to your starting point.  Based on quick calcs, your starting point would be about 1.16 miles north of the south pole, give or take.

• Anonymous
  December 05, 2008
  I think you can extend jnelso99's solution a bit further: Find the circle that has a one mile circumference around the Earth just above the South pole. Then start at any point one mile north of this circle. In that case, there are an infinite number of points from which you could go one mile south, go one mile east (which would take you on one complete trip around this circle), and then one mile north again to get back to your starting point. In fact, not only are there an infinite number of points one mile north of this circle, but there are also an infinite number of circles (you could be one mile north of the circle that has circumference of 1/2 mile, 1/3 mile, 1/4 mile above the South pole etc.) - going one mile east along any of these lines would simply take you several times around before ending back where you started on the circle, and then go one mile north again... At least, I think this works. I like these puzzles - keep them coming!

• Anonymous
  December 06, 2008
  Actually there are no polar bears at the north pole either. See http://en.wikipedia.org/wiki/Image:Polar_bear_range_map.png

• Anonymous
  December 08, 2008
  Last time , I posted a question sent to me by Rob Mount.  So what solutions exist other than the

• Anonymous
  December 25, 2008
  There are infinite number of such points in the south hemisphere :)

• Anonymous
  January 17, 2009
  I think hypothetically such situation can be created at various points on earth. For instance,   a situation where point lies on mountain top which is higher than 1 mile (basically cone shape where height of cone is more than 1 mile).   Then, walking 1 mile south is walking the mountain down 1 mile, then walking 1 mile east, which i again walking 1 mile on the circumference of mountain for a mile, and then walk 1 mile north, which is walking up 1 mile and you reach same point again.   The other solution can also be replicated using conical shape.   Such phenomenon is only possible on curved surfaces, and NOT plain geometry.

• Anonymous
  January 17, 2009
  The comment has been removed