The Locker Problem

My group and I found out that there were 31 perfect square roots in 1,000. Then we connected what we got to what other groups got and came out with the answer that when 1,000 students follow the pattern with the 1,000 lockers, there would be 31 lockers open. The lockers that would be open would be the ones that have perfect square roots. For example, 1, 4, 9, 16, 25, 36, 49, and until you get to 961, which its square root is 31. These lockers are open because only perfect square roots have numbers that multiply by themselves. For example, locker number 4 is open because it has been changed the state of, 3 times. It is 3 because you can make 4 with only three numbers, which are1, 2, and 4 because you don’t count the numbers twice. So 1*4 and 2*2. Only perfect square roots have odd amount of factors because they have numbers that multiply by themselves and you just count them once. Locker number 4 is open because student #1 opened it then student #2 closed it, then student #3 didn’t do anything to it, and at last student #4 opened it, so it stayed open like shown in the diagram below. Each factor corresponds to the student who touched the locker. The perfect square roots have to have odd amount of factors to be open because if they had even amount of factors they would be closed. Odd amount of factors mean open and even amount of factors mean closed.

Student #1:

The first student goes along the row and opens every locker.

Student #2:

The second student then goes along and shuts every other locker beginning with locker number 2.

Student #3:

The third student changes the state of every third locker beginning with locker number 3. (If the locker is open the student shuts it, and if the locker is closed the student opens it.)

Student #4:

The fourth student changes the state of every fourth locker beginning with number 4.

Locker #1:	1	2	3	4	5
Students that have opened or closed the locker:	1= Open	1=Open 2=Closed	1= Open 2= Closed	1= Open 2=Closed 4= Open	1= Open 5= Closed