A school hallway has a thousand lockers. A student goes down the hallway opening every locker. Then a second student goes down the hallway closing every other locker. A third student goes down the hallway, and at every third locker, he opens it if it was closed and closes it if it was open. This continues on for a total of 1000 students opening and closing lockers. How many lockers are open when the students are finished? 
 

Solution

The lockers which are open are the ones which have been "altered" an odd number of times. This means that their locker numbers have an odd number of divisors, which means they are all perfect squares. There are 31 perfect squares less than 1000. 

Isn't that slick?