One Mile South, One Mile East, One Mile North

Nowhere warm . The well-known point is the North Pole (what colour is the bear?).

Antarctica has no bears, but does have infinitely many solution points. There is a circle of latitude from which you can walk one mile south to end up exactly $\frac{1}{2\pi}$ miles from the South Pole. Your eastward path then winds around the South Pole exactly once; walking north then takes you back to your start point.

There are infinitely many points on this circle of latitude; but there are also latitudes from which your path circuits the South Pole twice, three times, etc.

The first solution that we might think of is the North Pole. That is because after one would travel one mile south from the North Pole, no matter how much that person travels east he or she would land back at the North Pole after traveling one mile north. This might seem like an obvious solution. However, are there any more such points?

Let's think of a point on Earth in the southern hemisphere at which one could walk one mile east and still arrive back at the same starting point. That is because that point belongs to the latitudinal circle (let's call it $c_1$ ) that's exactly a mile in circumference. Now that we have discovered this useful piece of information, we can imagine another latitudinal circle (let's call it $c_2$ ) that is exactly one mile north of our first latitudinal circle ( $c_1$ ). Any point on this circle ( $c_2$ ) will also be a solution because if one walks one mile south, that person will end up at our first latitudinal circle ( $c_1$ ). Then, walking first one mile east and then one mile north will bring that person to the starting point on ( $c_2$ ). Since ( $c_2$ ) consists of infinitely many points, we can see that the correct answer to the problem is infinity.

However, thinking further, we can see that for some circle ( $c_3$ ) that is $\frac{1}{2}$ miles in circumference, we can find more points that are also a solution to this problem. First, we have to notice that starting on any point on ( $c_3$ ) and traveling around ( $c_3$ ) twice would result in traveling one mile in total, and would also bring a person back to the starting point on ( $c_3$ ). Similarly, starting on any point on some circle ( $c_4$ ) that lays exactly one mile north form ( $c_3$ ) will also be a solution. This is because if one walks one mile south, then one mile east (twice around ( $c_3$ )), and then one mile north, that person would return to the starting point on ( $c_4$ ).

This pattern continues for any latitudinal circles that are $\frac{1}{n}$ miles in circumference, where $n=1, 2, 3, ...$ . This is because one can always travel $n$ times around such a circle and cover a mile while always coming back at the same spot. Just as before, any latitudinal circle that lays exactly one mile north from that one will contain infinitely many points that are a solution to this problem.

Therefore, not only that there are infinitely many points that are a solution to this problem, but there are also infinitely many latitudinal circles from which we can pick a point that will be a solution.

Lastly, try to explain why there are no such latitudinal circles in the northern hemisphere.