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Re: Where on Earth?

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On 7/18/2016 8:58 PM, pnalsing@gmail.com wrote:
> On Monday, July 18, 2016 at 6:28:15 PM UTC-7, Sergio wrote:
>> Where on Earth?
>> Where on the surface of the earth could you travel 1 mile south, 1 mile
>> east and one mile north and end up at your starting point?
>>
>> One answer is easy to find. But there are other answers. Many of them.
>> Where are they and how many are there?
>
> 1 at the north pole and infinity X infinity somewhere near the south pole...
>


The “harder” spots

The other spots on the earth all involve traveling near the South Pole.

The trick to these solutions is that you end up in the same spot after 
traveling one mile east.

How can that be?

One way this is possible is if you are on a line of latitude so close to 
the South Pole that the entire circle of latitude is exactly one mile 
around. We will label this circle C(1) for convenience.

With this circle in mind, it is possible to figure out a solution.

Let us begin the journey from a point exactly one mile north of C(1). 
Let’s trace out the path of going one mile south, one mile east, and one 
mile north again.

To begin, we travel one mile south to point on the circle C(1). Then, we 
travel east along the circle C(1), and by its construction, we end up 
exactly where we began. Now we travel one mile north, and we reach the 
starting point of the journey, exactly as we wanted.

he trip will look something like this rough sketch I made:

This demonstrates there is a solution involving a circle near the South 
Pole.

In fact, the circle C(1) is associated with a family of solutions. Any 
point one mile north of C(1) will be a possible solution. This means the 
entire circle of latitude one mile north of C(1) is a solution. This 
means there are an infinite number of solutions associated with the 
circle C(1)!

That alone seems remarkable. But what is more interesting is that there 
are even more solutions.

The circle C(1) was special because we traversed it exactly once, and 
ended where we started from, when we went one mile east.

There are other circles with the same property. Consider the circle 
C(1/2), a similarly defined circle of exactly 1/2 mile in circumference. 
Notice that traveling one mile east along this circle will also send us 
back to the starting point. The only difference is that we will have 
traversed the circle two times!

Thus we can construct solutions using the circle C(1/2). We start one 
mile north from C(1/2) and every point along this line of latitude is a 
solution. There is an infinite number of solutions associated with the 
circle C(1/2).

Naturally, we can extend this process to more circles. Consider the 
circle C(1/3), similarly defined with exactly 1/3 mile in circumference. 
It would be traversed three times if we travel one mile east along it, 
and we would end in the same place we started from. This circle too will 
have an infinite set of solutions–namely the line of latitude one mile 
north of it.

To generalize, we can construct an infinite number of such circles. We 
know the circles C(1), C(1/2), C(1/3), C(1/4), … C(1/n), … will be 
traversed exactly n times if we travel one mile east along them. And 
there are corresponding starting points on the lines of latitudes one 
mile north of each of these respective circles.

In summary, there are an infinite number of circles of latitudes, and 
each circle of latitude contains an infinite number of starting points.

The correct answer, therefore, is “an infinite number of circles of 
latitude near the South Pole, each containing an infinite number of 
starting points, plus one extra point for the North Pole.”

Did you figure it out?

This problem appears in my book Math Puzzles Volume 1 (nearly 5 star 
rating) which contains many interesting brain teasers and math riddles.

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Where on Earth? Sergio <invalid@invalid.com> - 2016-07-18 20:28 -0500
  Re: Where on Earth? pnalsing@gmail.com - 2016-07-18 18:58 -0700
    Re: Where on Earth? Sergio <invalid@invalid.com> - 2016-07-18 23:43 -0500
      Re: Where on Earth? noTthaTguY <abu.kuanysh05@gmail.com> - 2016-07-20 15:27 -0700

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