In considering the locker problem, I've concluded that, once all 1000 students go through, 31 lockers will be closed. All other lockers will be open. The lockers that I claim will be closed are the perfect squares that are less than (or equal to) 1000, i.e., lockers 1, 4, 9, 16, ..., 900, and 961.
Your visual layout is great — the way you map out the first few cases and tie your conclusion to the number of factors shows really clear mathematical reasoning. I also love that you identified the perfect-square pattern cleanly and checked it against 1000.
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