1000 Lockers Problem

The problem is a very interesting one. It did confuse me as which way to go at the beginning. I think for me, the most difficult part is how to establish the numbers of lockers and the students in a correct format so that we can record the state of the lockers to show a clear pattern. After many attempts, I finally set it up as the pictures show. There must be a pattern and I did try the first 10 combinations, it turns out the closed lockers are 1,4, 9, and the opened lockers are 2,3,5,6,7,10. I realized that 1,4, and 9 are actually perfect squares for 1,2, and 3, but will this pattern be continued until 1000? I later on tried more numbers (up to 20) but only found one more perfect square which is16. But it is sufficient enough to make the conclusion: if the number of the lockers are perfect squares, then they will be closed at the end; if the numbers of the lockers are not perfect squares they will be open at the end. When I searched on the internet, it explained in more details. The solutions also suggested that perfect squares have odd number of factors and non perfect squares have even number of factors which determines the state of the lockers. However, I personally feel that perfect squares are easier to detect.