Locker Problem

The way I got the answer was that after doing it through 16 times I found that a pattern emerged. This pattern shows that out of 1000 lockers there are only 31 of them that remain open. The 31 lockers are made up of all perfect squares. The perfect squares are all unique compared to the rest of the lockers. All their factors add up to being odd. For example, 16 is made up of one times 16, four times four, and eight times two; As you can see it has a odd number of factors instead of the usual equal amount of factors. It seemed to me that every time you go through the odd ones would get skipped the next time around leaving them open as my diagram below clearly shows.