The Locker Problem

          We were given a problem in math. There are 1,000 lockers, each numbered. Student #1 opens every locker. Student #2 changes the state (if it's open, it closes. If it's closed, it opens) of every 2 lockers. Student #3 changes the state of every 3 lockers, and so on until the 1,000th student goes. To solve this, we used two-sided coloured counters, with a different colour on each side.. In the textbook, it said that you could use only 30 counters to save time. 

          First, we started with 30 counters on the red side, representing closed lockers. Then, we changed the state of every one by flipping them over to the yellow side representing open lockers, because Student #1 changes every one. Now all of the lockers were open. Next, we changed the state of every other locker because Student #2 changes every other locker. Now, all of the even number lockers were closed. All of the lockers that were a multiple of 2 (lockers that were numbers that 2 was a factor of) were changed. Then we changed the state (flipped) of every third locker because Student #3 changes the state of every third locker. We continued this way until we got to the 30th student.

          Sadly, we messed up our flipping at about the 17th locker, but we had gotten just far enough for me to notice a pattern. The open lockers were lockers 1, 4, 9, 16, and 25. All square numbers. 

          Square numbers have an odd number of factors no matter what. Square numbers are all products of a number multiplied by itself. That means that one of their factors, because it is multiplied by itself, has no pair, that make the number or factors odd. For example, 25's factors are 1, 5, and 25. There are three factors because 5 is multiplied by itself. 

          The reason the square number lockers are left open is because of the number they are changed. If the locker is changed an even amount of times, it will always end up closed. For example, if a locker is changed once, it'll be open. If a locker is changed twice, it'll be opened and then closed again. Each locker is opened when the number student that is a factor of the locker number goes. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. When each of those students go, the locker will be changed.

Number Locker	Students who Changed it / Factors
1	1
2	1, 2
3	1, 3
4	1, 2, 4
5	1, 5
6	1, 2, 3, 6
7	1, 7
8	1, 2, 4, 8
9	1, 3, 9
10	1, 2, 5, 10
11	1, 11
12	1, 2, 3, 4, 6, 12
13	1, 13
14	1, 2, 7, 14
15	1, 3, 5, 15
16	1, 2, 4, 8, 16
17	1, 17
18	1, 2, 3, 6, 9, 18
19	1, 19
20	1, 2, 4, 5, 10, 20
21	1, 3, 7, 21
22	1, 2, 11, 22
23	1, 23
24	1, 2, 3, 4, 6, 8, 6, 8, 12, 24
25	1, 5, 25
26	1, 2, 13, 26
27	1, 3, 9, 27
28	1, 2, 4, 7, 14, 28
29	1, 29
30	1, 2, 3, 5, 6, 10, 15, 30

Lockers left open-
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.