I was pleased that I had not yet met "The Locker Problem". My first step was to rewrite the puzzle in my own words so that I could work on it in my notebook. From there I broke the problem into manageable chunks of N = 1, 2, 3, 4, 5 students to see if any patterns emerged. I noticed that once the Nth locker has changed states from the Nth student, it no longer changes states because no other students will be eligible to touch it. I decided to investigate the idea of N with odd factors because an uneven number of state changes results in a final state of closed.
I checked to see if there are any square numbers with even number of factors. By nature of square numbers, they have an odd number of unique factors.
Thus, I concluded that perfect squares 1, 4, 9, 16, ... 961 are the doors that are closed after the 1000 students are done their shenanigans.
It has been too long since I've been captivated by a math puzzle. I
loved having them in my classrooms throughout my high school education.
However, in university I often had to go out of my way to get lost in
thought with puzzles. I will be accumulating my favorite puzzles so that
I can provide my future students with endless scribbles in the margins
of their notebooks.
Nice work here.
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