Solution:

Did you consider the southern hemisphere? If you did you must be knowing that there is a ring near the South Pole with a circumference of one mile. Consider the situation that you are standing on one mile north of the ring at any point. Now if you walk on the southern direction and cover one mile, you will be standing on the ring. Traveling one mile east will bring you on the circumference of the ring. Walking one mile north from there, you will be standing on the exact point where you started. Now if you start counting, you will understand that while you walk 1 mile north in the end, you can reach an infinite number of points.

In such a case, the total number of possible points possible are 1 + infinite.

Now let us consider the ring that is half a mile in circumference near the South Pole. If you walk a mile along the ring, you would circle twice but will reach the point where you started from. In such a case if you start from the point that is located one mile north of a half mile ring, it will also help you reach the starting point after traveling as per asked.

Now with every possible integer N, there is a circle with radius R = 1 / 2 (2*pi*n); which is centered at the South Pole.

If you walk along these rings, you will be circling N times and again returning to the point where you started. You must note that the possible values for N are infinite. Also, you can have infinite ways of selecting a starting point which is located one mile north of the rings which means that there are (infinite * infinite) possible points.

Concluding with our statements, the possible number of points are equal to 1 + infinite * infinite which is equal to infinite.

Thus there are infinite points possible.