On the first day back after Spring Break, the students enter a long hallway containing one thousand lockers. The first student, as he walks from one end to the other, opens every locker door. The second student, as she follows the first, closes every other door. The third student follows the second, and switches every third door; that is, if a door is open, he closes it, and if a door is closed, he opens it.
The students continue with this pattern, the n-th student switching every n doors. After a thousand students have walked down the hallways of lockers, which doors end up being open?
Please tell me there is a simple, or simple-ish, way of doing this!?!