The Locker Problem (1000 lockers edition)

Before attempting the problem, I thought the solution would have been a messy array of numbers that needed to be tracked via extensive diagrams. However, I began the problem by setting up 10 lockers, perhaps due to our connection to the base 10 system. Very quickly, it seemed that base 60 would be great for a more obvious illustration of the pattern since it has significantly more factors than 10. That level of effort may be commended, but would end up redundant. 

To begin, my diagram has a black underline beneath each of the changed lockers. The rows indicate the changes made by a Person, the columns indicate the states of the locker each round. 

After the 10th iteration, I noticed a few things: Locker 1 remains closed. Locker 2 is changed by Person 2, and remains open. Locker 3 is changed by Person 3, and remains open. Locker 4 is changed by Person 2 and Person 4, and ends up closed. Locker 8 is changed by Person 2, Person 4, and Person 8. What's interesting about Locker 8, is that Person 1 closes, Person 2 opens, Person 4 closes, and Person 8 opens. Locker 8 should remain open, because no one else after Person 8 will touch it. Similarly, with Locker 9, Person 1 Person 3 and Person 9 interact with it, but it remains closed, just like Locker 4. 

Now why does Locker 8 stay open, but Locker 9 stays closed? The solution to the locker problem lies in this observation.  Locker 8 has an even number of changes, while Locker 9 has an odd number. From this, we confirm that because 8 has an even number of factors, while 9 doesn't. 

The next question is, what types of numbers have an odd amount of factors? Number that have factors like: 1, a, b, c, d, e, N. c is multiplied by itself, making N a square. 

This looks like we have all the part necessary to solve the locker problem, even though we did just 10 iterations. On to the solution!

SOLUTION: The lockers that are closed after the 1000th student has come through are the squares- Lockers 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, and 961.

Here's the diagram that guided my thought-process.