Math Puzzle

When looking at the problem, I first looked at the pattern by writing out whether each locker is closed or open.

I wrote down a diagram of ten lockers and based on the result, I can see that the first locker, fourth locker and the ninth locker are closed. 

I then dug deeper by writing out numbers up to twenty and I found that all the lockers whose numbers are perfect square are closed while the non perfect square number lockers are open. Not only that, I also found that the perfect squares have an odd number of factors. For example, 4 has three factors:1, 2 and 4. The next perfect square 9 also has three factors: 1, 3 and 9. On the other hand, non-perfect squares have an even number of factors, such as 6 has four factors: 1, 2, 3 and 6 while 12 has six factors: 1, 2, 3, 4, 6 and 12. 

Therefore, we can conclude that numbers with an odd number of factors are closed, which are the perfect squares in this case, while non-perfect squares remain open.

Set of lockers:

1st closed (1 factor)

2nd open (2 factors)

3rd open (2 factors)

4th closed (3 factors)

5th open (2 factors)

6th open (4 factors)

7th open (2 factors)

8th open (4 factors)

9th closed (3 factors)

10th open (4 factors)

11th open (2 factors)

12th open (6 factors)

13th open (2 factors)

14th open (4 factors)

15th open (4 factors)

16th closed (5 factors)

17th open (2 factors)

18th open (6 factors) 

19th open (2 factors)

20th open (6 factors)

Not only that, I also noticed that the number of open lockers is increasing by two between each set of closed lockers.

To prove this, I tried writing out an equation for this result. Let's say 2 x n, where n represents the position of the set of open lockers, the first set of open lockers is two because we know 2 x 1 is qual to 2, and we can attest that by seeing there are two open lockers between the first two locked numbers. The next number n = 2, which represents the second set of open lockers, we plug it into our equation 2 x n and we find that it is equal to four, so we know that there are four open lockers between the next two locked numbers and so on. By doing this, we can also see that each locked numbers we end up with are perfect squares.

I can safely assume that for lockers up to 1000, every perfect square lockers are closed while every other lock numbers are open.