Teacher bird here.
I'm not sure why, but when I first approached this problem I was expecting primes to play a role. Not sure why this was, but I initially thought both primes and squares would be open, until I saw that 5 was closed, and realized I had no justification for primes being open...
Initially I wrote out a sequence of 10 to get a feel for things. I only wrote out the changes in position, and then a final line which showed the position of the first 10 lockers after 10 students. I didn't notice that it was only the squares at this point, but had a feeling it was related to the factors of numbers. I wrote out the factors for 60 and was reminded that they come in pairs. This helped me realize that every locker would be closed except for those with an odd number of factors. Since only squares have an odd number of factors, the problem was solved.
I love this question, though I'm not sure if the ceiling is as high as I originally thought. Once someone understands the algorithm, all higher cases are trivial. This might be good for a Math 9 class. If was to present this to a Math 9 class, I would ask questions in this order:
Given the problem, how many lockers are open if there are:
a) 10 Lockers
b) 50 Lockers
c) 100 Lockers
d) N Lockers
I would make sure to only show them one part of the question at a time so that they are forced to realize brute force will not work because of their own experience with the question, NOT because they are anticipating a harder case later on.
Thanks Jacob! A question to ask yourself: why did you select multiples of 10 lockers to try? (10, 50, 100) Was that especially helpful -- and would it be helpful to your students? Any other choices that might work well (like, say, 25..??)
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