Due to tiredness and a general difficulty with visualization (I am 99% sure I have aphantasia), I skipped the "classic" 1st step of trying it out in my head and went straight to brute-forcing the n=10 case and looking for patterns:
Through this process, I found that if the number of the locker has an even number of factors it will be closed at the end, and if it has an odd number of factors it will be open at the end (excluding 1 as a factor in both cases for some reason 😅).
I proceeded to ask how many lockers will be open at the end of the process and tried to build a series using the pattern that each "group" of open lockers is double the size of the previous. I struggled to find a good way to iterate & bound the series though and eventually gave up.
I then challenged my partner (also a math teacher) to try the problem:
She found that the number of each closed locker is a perfect square, something I just didn't see 😮.
From this, determining the number of open and closed lockers became easy because I just needed to find the largest perfect square that is less than 1000.
Starting from 30^2=900, I went up and found 31^2=961 & 32^2=1024 (was too lazy to get a calculator so I did it by hand). This told me that there were 31 closed lockers and therefore 969 open lockers.
Sometimes, students don't feel comfortable asking for help as you did for this problem. What strategies will you try to encourage students to (1) persevere even when the math becomes challenging and, (2) to ask for help when they become stuck? Will you encourage the use of online resources or collaboration with their peers?
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