Reply #19: The Answer is: [View All]

[DustMolecule]

One thing to 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
pattern?

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:

this Student leaves locker 48

1 open
2 shut
3 open
4 shut
6 open
8 shut
12 open
16 shut
24 open
48 shut

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 (without having to work it all out!).

Here's a link with some visuals:
http://mathforum.org/alejandre/frisbie/locker.look.html

You can also work it out in a spread sheet:
http://mathforum.org/alejandre/frisbie/student.locker.html