The Locker Problem

It is a pretty fun problem! So just like all problems, I would write down what I know, so in the case, just the question itself. We know that after the first student, all lockers are open, so I drew out the locker configuration, with open squares being open lockers and shaded squares being closed lockers. My first attempt is in blue ink, where I only did 5 lockers and 5 students. I quickly realized that after the fifth student, the locker configuration remains the same, since the first locker that the sixth student touches is the sixth locker. This means that whatever the student is, the configuration before the student number is locked.

So for my second attempt, I did 10 lockers and 10 students, and I found that at the tenth student, the first 9 lockers on fixed for the rest of the riddle. The lockers that are open are 1, 4, and 9, perfect squares. So knowing this, I found the largest perfect square below 1,000. I know 30 squared is 900, so I started there. And when I tried 31 squared, I knew whatever perfect square after 961 would be bigger than 1,000. So, I concluded that the lockers that remained open are 31.

So the primary mathematical thinking I did was mainly just drawing things out and finding patterns! I can definitely see myself doing this as a bonus question for my exams hahaha :)