Entrance Slip #2: The Locker Problem

 When I first began this problem, I tried to parse the question. I needed to visualize what was going on so I draw out the lockers as squares with open squares representing open lockers and closed squares representing closed lockers. I drew out the status of the lockers after the first 10 students had gone through and made their changes. From there, I tried to find a pattern. At first, I thought it might have something to do with the difference of closed lockers at the end of each student's turn but moved away from that thought when I realized that lockers 501-1000 could only have one locker closed at a time (since the 501st student would only alter the status of 501, the 502nd would only alter the status of the 502nd locker etc.). However, I still couldn't tell what the status of the locker would be at these stages so I decided to take a closer look at what was happening for the first 10 cases:

I realized that all primes were open, and from there I realized that the status of the door (either open or closed) corresponded with how many factors the number had. For example, 10 was open because it has an even number of factors (1, 2, 5, 10) while 4 was closed because it had an odd number of factors (1, 2, 4). I wasn't able to see the full pattern right away so I used the property (# of factors of m) = (n_0 + 1)*...*(n_f + 1) for p^(n_0)*...*p^(n_f) = m, where m equals the number (also, if anyone knows how get sub/superscript on blogs, please let me know). Through this process, I realized all square numbered lockers are closed and the rest are open after all 1000 students go through and make their alterations (the specific square numbers are listed in the picture below):