This article is from the Puzzles FAQ, by Chris Cole email@example.com and Matthew Daly firstname.lastname@example.org with numerous contributions by others.
How far must one travel to reach the North Pole if one starts from the
equator and always heads northwest?
One can't reach the North Pole by going northwest. If one could, then
one could leave the North Pole by going southeast. But from the Pole
all directions are due south.
If one heads northwest continuously, one will spiral closer and closer
to the North Pole, until finally one can't turn that sharply.