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I've seen a variation of the 2nd puzzle where all the students change the state of the locker, instead of it being a rota of open, close, change, like it is here.
I am beginning to think you are right about perfect squares having an odd number of factors, so doors open because the square root factor is only counted once, and all the other factors are paired up. Nice work!
I really like the locker question. I am so tempted just to program it and get a result, but that's kind of cheating isn't it. I could show the results graphically after each student inorder to learn.
1. Since we want to show a remainder when dividing by 5, we must first get something that is divisble by 5. So lets consider the 5 cases:
1) Prove that no square integer number can have a remainder of 3 when divided by 5