I got this problem from Rustan Leino, who got it from Carroll Morgan.
I solved it and wrote up my solution.
A building has 16 rooms, arranged in a 4x4 grid. There's a door between every pair of adjacent rooms ("adjacent" meaning north, south, west, and east, but no diagonals). Only the room in the northeast corner has a door that leads out of the building.
In the initial configuration, there's one person in each room. The person in the southwest corner is a psycho killer, and the rest cannot move because they're chained to the floor. The psycho killer has the following traits: If he enters a room where there's another person, he immediately kills that person. But, he also can't stand the sight of blood, so he won't enter any room where there's a dead person.
As it happened, from that initial configuration, the psycho killer managed to get out of the building after killing all the other 15 people. What path could he have taken?
There are many possible paths, but they all start the same way: after killing the first person, he must return to the room where he started; he can do so because he never killed anyone there. For example, one possible complete path is: north, south (back to the room where he started), east, north, north, west, north, east, east, south, south, south, east, north, north, north, exit.