The Locker Problem

In this problem, we had to find out how many lockers would be open if there were 1000 lockers, and one person opens all of the lockers, the 2nd person closes every other locker, the 3rd person changes the state of every 3rd locker, and so on.

First, we tried to find a pattern that we could build from. We messed with bottle caps to see if we could find one, and lo and behold, we did. We found out that there was one open locker, then 2 closed lockers, then another open locker, then 4 closed lockers, then another open locker, then 6 closed lockers and on and on. From this, we found yet another pattern that would help us even more.

We continued going through numbers, seeing if there was a better way to do this, and we found out that all the open lockers are perfect squares, being the square root of the number would be a whole number without and decimals (ex. 9, 16, and 25). There are 31 If you look at the attached diagram, you’ll see why that the perfect squares are the only lockers that are open. The diagram also almost explains why perfect squares have an odd number of factors, but not quite.

Perfect squares have an odd number of factors simply because one of the factors is there twice, but only counts once. For example, in 47, you can multiply 7 by 7, but only one 7 would count as a factor.