The Locker Problem!

There were 1,000 lockers. One student opens every locker, another student closes every other locker. Leaving 500 lockers open, and 500 lockers closed. The diagram below shows the fist 16 lcokers, and the pattern. I discovered that all perfect sauared lockers have factors that are pairs, so now every locker except the perfect squared lockers get opened, and closed an even amount of times. There are now 31 lockers open, every locker number that is a perfect square is open like, 1, 4, 9, 16, 25, 36, 49, 64 etc... Thos Lockers are open because they are perfect squares.

X=Closed O=Open

XOOXOOOOXOOOOOOOX