The Locker Problem

There are 1000 new lockers at school. One student goes and opens every single locker. The next student changes the state of every other locker, starting with number two. The third student changes every three, the fourth every four, and so on until one-thousand students have changed the lockers. How many are open still? Why? What is the pattern?

- - - - - - - - - - - - - - - - - - - - - - - - - - -

There are 31 doors open, and they are all the perfect squares, such as 1, 4, 9, etc. That is because while all non-perfect squares were changed an even amount of times, with an even amount of factors, the perfect squares were changed an odd number of times, being that they have an odd number of factors. This means that the pattern is as follows:

|o| |c| |c| |o| |c| |c| |c| |c| |o| |c| |c| |c| |c| |c| |c| |o| |c| |c| |c| |c| |c| |c| |c| |c| |o|...

(|o| = open locker |c| = closed locker)