The Locker Problem

Coco S.

The Locker Problem

There are 31 out of 1000 lockers that are open. The ones that are open are perfect squares, as in they use square roots to find the answer. ie: 2x2=4, 3x3=9, 4x4=16. There are 31 out of 1000 lockers that are open. The ones that are open have an odd number of factors, because the square root is its own factor. They have an odd number of factors because when you touch it an even amount of times it'll just end up being closed. So if you touch a locker two times, it'll be opened, then closed. If you touched it three times, it would open, close, and open. The lockers that are open are 1, 4, 9, 16, 25, 36, etc. We can continue to find out which ones are open by finding the square roots. So every kid that goes and changes the state of every other locker or every two lockers, eventually the kids would be skipping all the ones closer to the beginning. Thus never being changed again.
In my diagram below you can see that lockers 1, 4, 9, and 16 are open. They are open because they have an odd number of factors.


Factors:

Locker 1: 1x1
Perfect square
Locker 2: 1x2
Prime
Locker 3: 1x3
Prime
Locker 4: 1x4, 2x2
Perfect square
Locker 5: 1x5
Prime
Locker 6: 1x6
Prime
Locker 7: 1x7
Prime
Locker 8: 1x8
Prime
Locker 9: 1x9, 3x3
Perfect square