One thing we can do is to let the first $10$ students go do their open/
shut thing with the lockers. The students who come after them are not
going to touch lockers $1-10$, so we can see which ones in that first
batch are still open and try to guess the pattern.
When we do that, we find that lockers $1, 4,$ and $9$ are open and the
others are closed. Now, that isn't much to go on, so maybe you could
let the next $10$ students go do their thing. Then the first $20$ lockers
are through being touched, and we find that lockers $1, 4, 9,$ and $16$
are the only ones in the first $20$ that are still open. So what is the
Let's take any old locker, like $48$ for example. It gets its state
altered once for every student whose number in line is an exact
divisor of $48$. Here is a chart of what I mean:
Student number leaves locker 48
Notice that $48$ has an even number (ten) of divisors, namely
$1,2,3,4,6,8,12,16,24,48$. So the locker goes open-shut-open-shut ...
and ends up shut. Any locker number that has an even number of
divisors will end up shut.
Which numbers have an odd number of divisors? That's the answer to
this problem. Just to help you along, here are the locker numbers up
to $100$ that are left open:$$1,4,9,16,25,36,49,64,81,100.$$
See if you can describe these numbers in a different way from "having
an odd number of divisors." Think about multiplying numbers together. When you understand how to describe them, you will see that $31$ of the
$1000$ lockers are still open.