Anyone here heard of the Five-room puzzle?
A rectangular building is divided into five rooms, with a door in each wall, as above. Is it possible to go through each wall exactly once?
. . . SPOILER AHEAD! . . .
Well, it isn't in a regular world. http://en.wikipedia.org/wiki/Five_room_puzzle#Solutions has a proof.
However, in a toroidal world, it is indeed possible!
Here's an animation showing both cases...