Locker Problem

When I first looked at this problem, I had a lot of ideas rolling around in my head. I started off thinking about multiples, but that was only a quick thought as I thought there might be a better approach. I realized early that person 501 and onwards would only touch one locker, so I would only need to figure out the first half of the problem and then flip the last half of the lockers.

I decided to work on the first 10 lockers and see if I could spot a pattern.

1:cccccccccc

2:cococococo

3:cooocccooo

4:coocccccoo

5:coococccoc

10:coocooooco

From this, I started to see a pattern that 1, 4 and 9 were all closed. I wanted to try 20 to see if my suspicions were correct and that the perfect squares were all closed.

I got 20: coocoooocoooooocoooo

From there I saw that 16 was also closed, and this was enough for me to say that all the perfect squares are closed, and all others are open.
I thought about this and wondered what makes perfect squares have this property that all of them would be open. I thought back to when I was teaching math last year and how we went over perfect squares and one of the interesting things is that they have an odd number of factors. Every other number will have an even number of factors, so they will all be touched an even number of times, so they will go back to their starting positions. Perfect squares will be touched an odd number of times, so they will change from their starting position.