Locker Problem 2nd Draft

After testing it out, I've concluded that there will be 31 lockers open in the end. The reason for that is simply because there are 31 perfect squares from 1 - 1,000. Why are the lockers who have perfect squares open? Well, the perfect square lockers have been touched an odd number of times because there is an odd amount of factors for each perfect square. For example; the factors of 9 are 1, 3, and 9. Since each locker was initially open, in order for it to be open again, it must be touched an odd number of times.