One mile south, one mile east, one mile north – a classic puzzle

I was having dinner with a friend the other day. We were talking about my writing, and out of the blue he asked me what my favorite puzzle was.

A lot of the puzzles I wrote about on this site came to mind, including

The hat puzzle

A mystery Sherlock Holmes couldn’t solve, but you can

The problem of finding true love

I was going to bring up one of these puzzles. The problem was that they were hard to explain over dinner.

I thought for a second about a puzzle that would be a little bit easier to state and explain.

And that’s when a classic problem came to mind.

I read about this first in the fun puzzle book How Would You Move Mount Fuji?. Years ago, Microsoft apparently used to ask this puzzle as an interview question.

Here is the problem:

How many points are there on the earth where you could travel one mile south, then one mile east, then one mile north and end up in the same spot you started?

My friend gave it some thought.

After a moment, he replied the answer was 1.

I told him he was on the right track, but his answer was incomplete.

Can you figure out how many points there are?

The solution is quite interesting. Answer below.
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The “easy” answer

The place my friend was thinking about was the North Pole.

This is, in fact, one of the correct spots.

You can trace out the path on a globe. From the north pole, you can move your finger south one mile. From there, you will go east one mile and move along a line of latitude that is exactly one mile away from the north pole. You finally travel one mile north, and you will exactly end up in the north pole.

The route you travel will look like a triangle or a piece of pie, as seen in this rough sketch I made:

This is one correct answer. But it is not the only one.

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.

The 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?