The Locker Problem - my notes/solution

I intuitively begin these sorts of problems by considering a smaller case, as this will likely reveal patterns that apply to the larger quantity in the question. I also tried writing out the same 10 case in different ways, as my first attempt did not capture/reveal the pattern as clearly as the second more visual attempt. In my notes I wrote out the case for both 5 lockers/students, as well as 10. The 5 case revealed little, and so expanding to 10 was necessary to find a possible pattern. When I saw that 1 , 4 and 9 were the only ones closed in the 10 case, I asked myself what these numbers had in common and immediately theorized that the pattern was locker numbers which aligned with perfect squares would be the only ones closed by the end of the procedure. 

However, this was technically only conjecture. I then had to consider an explanation for why perfect squares would essentially be "modified," while all other numbers of lockers would just end up open again, back in their original state. To return back to an original state would require an even number of students acting upon it, such that for each student that closes a locker, there is another that un-does this action. In essence, this meant all the students could be paired up as one doing and one un-doing. From here I realized that the perfect squares would not have this even number of students to undo the last closing, as one student (aligning with the number's square root) would act on the locker's state, without another student (a factor pair student) to undo their action.