The Locker Problem

I first solved a problem similar to this a couple years ago. The problem I was given involved 100 lightbulbs and I initially just went lightbulb by lightbulb and seeing whether each was on or off. I remember being too lazy to get out a piece of paper so I just did it in my head. By the time I got to around locker 20, I saw a pattern - all lightbulbs in the positions of perfect squares were on. I realized that this was because the number of times a lightbulb was switched on or off was equal to the number of factors it had and only perfect squares had an odd number of factors (all factors of a number have a corresponding factor which when multiplied by each other equal the number except the square root of a perfect square).

I realize that a student attempting my method may not see the pattern as quickly as I did and may eventually get frustrated and give up. I also imagine that other methods might be more popular. I have seen many listing out the lockers and instead of going locker by locker, they went student by student changing the status of each locker as each student went by it. If a student had this method in mind, the sheer number of lockers might be daunting. If i approached a student who told me that they were having trouble with the number of lockers, I would suggest that they focus on maybe the first 30 lockers and see if they notice anything. Completing this task would hopefully allow them to see a pattern and then hopefully give them the opportunity to think about why that pattern is the answer.

As an extension, you could specify various lockers that you wanted open and then ask for which of the 1000 numbered students should be dispatched to open/close their respective lockers.