Answer: Tuesday Teaser #19
The textbook answer to this is to name the North Pole as the only place where it can be done. Indeed, it is definitely true that the North Pole is the only place where you can walk a mile south (to a point A, say), then a mile east (to a point B, say) and then a mile north, arriving back where you started, such that point A is not equal to point B. However, what if point A is equal to point B? Imagine drawing a line around the Earth (close to the South Pole) that is a mile in circumference. If you were to start anywhere on the line and walk a mile east you would arrive back where you started. Such a line definitely exists, so we can conclude that if we start at any point a mile north of the line, we obtain another place with the property that we want. This gives us an (uncountably) infinite number places on Earth that have the property.
We can even go one further than this: instead of walking once around the line and arriving back where we started, we can draw lines 1/2 mile, 1/3 mile, 1/4 mile, etc. long and walk around them 2, 3, 4 times to arrive back to where we started.
In conclusion, there are infinitely many points on Earth from which we can walk a mile south, a mile east and a mile north and arrive back at the place where we started.
