This is interesting question, I would like to solve in my way.

Yes,it contains infinite number of solutions.It is depends on the number of points we take on earth.

Now let us imagine that earth is a perfect sphere,one spot that the answer is north pole.If you Starting at north pole when we go 10 mile south then 10 mile west and finally 10 mile north we are going on a triangular path exactly to north pole. This is one of the answer to the puzzle.This is what we can see in below diagram.

In fact , there are many more answers to this puzzle.Now let us think about a perfectly spherical earth. Now let some place near to south pole.Imagine a circle exactly one mile around in circumference very near to south pole,we cal it as circle C(1).Now, imagine 10 mile south from circle C(1) then 10 mile west and finally 10 mile north then you finally end up with starting point. So,this is another type of solution to this puzzle.

The circle C(1) was special because we traversed it exactly once, and ended where we started from, when we went 10 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 10 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 10 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 10 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 10 mile east along them. And there are corresponding starting points on the lines of latitudes 10 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.”