342 Locker Problem Solution Sep 14

 

Here are my notes on how to solve the locker problem.

I realized first that if a locker is touched an even number of times it will end up in the same state as the beginning, open. If it is touched an odd number of times it will end up closed.

Each locker number has a certain number of factors, lets use the variable n for the number of factors. For example, locker number 20 has the factors: 1, 2, 4, 5, 10 and 20. That is a total of 6 factors so n=6. Since 6 is even, the locker will end up open.

This approah can be used to determine how many times each locker is touched and therefore whether it will end up open or closed.

Update: I just realized (many hours later) that the only closed lockers are the ones that are perfect squares like 4, 9, 16, 25. All the open lockers have an even number of factors because the factors come in pairs. That is, unless it is a perfect square. In this case, one factor is a square root which has no pair because it is it's own pair.