Solution for #8

Alternate Solution #1

Assuming a perfectly spherical Earth, somewhere one mile north of the latitude (in the southern hemisphere) that is one mile in circumference. The man walks south one mile to this latitude and walks one mile east, which takes him all the way around and back to where he started. The last step (one mile north) retraces the first step he took (one mile south).

Alternate Solution #2

The man could also be one mile north of the latitude that is one half mile in circumference -- on second leg of the journey, he'd go around twice instead of once. He could also be one mile north of the latitude that is one third of a mile in circumference, or one mile north of the latitude that is one quarter of a mile in circumference, and so on.