The 1000 locker problem

When attempting to solve this problem, I began by rewording the information provided. I then drew pictures to help visualize the problem. I defined ‘O’ to be an open locker and ‘X’ to be a closed locker. I also defined student 0 to be the initial state of the lockers before student 1 closes them. After drawing locker changes up to student 4, and extending the lockers altered to locker 15, I began to write out patterns I saw. For example:

I saw that locker 2 was opened 1000 times (student 0, 2, 3, …, 1000) and closed once (student 1).

I saw that locker 4 was open twice (student 0 and student 2) and closed everywhere else (student 1, 3, 4, 5, …, 1000) for a total of 999 times.

I saw that locker 9 was opened 7 times (student 0, 3, 4, 5, 6, 7, 8) and closed 994 times (student 1, 2, 9, 10, 11, …, 1000).

I saw that locker 15 was opened 989 times (student 0, 3, 4, 15, 16, 17, …, 1000) and closed 12 times (student 1, 2, 5, 6, 7, …, 14).

From here, I saw a pattern and came up with the conjecture which states: “lockers opened an odd number of times will be open at the end [after the 1000^th student].” At the time of working on the problem, I wanted to conduct a few more tests to verify the conjecture:

Consider locker 18. It was opened 988 times (student 0, 2, 6, 7, 8, 18, 19, …, 1000) and closed 13 times (student 1, 3, 4, 5, 9, 10, 11, 12, … ,17). 

This counterexample showed that the conjecture was false as locker 18 was opened an even number of times and remains open after the 1000^th student. As I am writing this blog now, I immediately see that this conjecture is false by the counterexample of locker 9 as the locker was opened an odd number of times but remains closed.

I then noticed that factors of the locker numbers were related to whether they were opened or closed, so I began listing out factors. I discovered that the lockers with an even number of factors would be open and the lockers with an odd number of factors would be closed. The only time a number would have an odd number of factors is if it was a perfect square. As the first 31 perfect squares are below 1000, that led me to conclude that there would be 31 lockers closed and 969 lockers open after all 1000 students had gone through opening and closing.