1,000 locker problem

When initially presented with the 1,000 locker problem, my immediate strategy was to scale the scenario down to a manageable size. Rather than attempting to conceptualize all 1,000 lockers at once, I mapped out just the first 20 lockers. By manually tracking the open and closed states as the first 20 students passed through, I created a low-stakes environment to test my conjectures and observe emerging patterns.

My primary mathematical thinking relied on inductive reasoning and analyzing factors. After noticing a slight miscalculation in my initial count (I originally thought 15 lockers would be open instead of 16), I paused to consider the core mechanism at play: a locker's state is determined entirely by the number of its factors. Lockers with an even number of factors receive an even number of "touches" and remain open, while those with an odd number of factors receive an odd number of touches and end up closed.

Connecting this observation to the properties of perfect squares, the only integers that possess an odd number of factors because one factor pairs with itself, was a pivotal shift in my approach. It allowed me to simply calculate the largest perfect square under 1,000 ($31*31=961$) to find the final answer.

I have documented my initial scratch work, including the wrong turn I took before realizing my counting error, and photos of these diagrams and factor patterns can be seen attached to my blog post. This was a fun experience and an excellent reminder that we must allow students the space to test small conjectures so their independent mathematical reasoning can firmly take root.

Parminder Kaur