Okay, I figured out #11:

"Other than the North Pole, where on this planet is it possible to walk one mile due south, one mile due east and one mile due north and end up exactly where you began?"

There are lots of possible answers (an unlimited number actually).

* Consider the line of latitude near the south pole where the circumference of the earth is exactly one mile. You start at any point that is exactly one mile due north any point on this line of latitude. So, following the instrucitons: (i) you walk one mile due south, and end up on this line at a point we'll call p; (ii) you walk one mile due east, and since the circumference of the earth at this latitude is exactly one mile, you end up at p again; (iii) you walk one mile due north and end up exactly where you started.

* There is a line of latitude near the south pole where the circumference of the earth is exactly half of one mile. Again, start at one point due north of this line. Walk one mile south, and up at a point on the line we'll call p. Now, in walking east one mile you pass point p at half a mile, and end up on point p at the completion of the mile. Walk one mile north and end up where you started.

* Continuing this process, you can start at any point one mile due north of any point on a line of latitude near the south pole where the circumference of the earth is exactly 1/n miles, where n is a positive integer.