Brain Teaser - The Locker Room

[DustMolecule]

At a new high school, there are exactly 1,000 students and 1,000 lockers. The lockers are numbered in order from 1 - 1,000.

On April Fool's Day, the students played the following prank:

The 1st student to enter the building opened every locker.
The 2nd student closed every even-numbered locker.
The 3rd student 'changed' (if opened, they would close it and vice-versa) every 3rd locker.
the 4th student 'changed' every 4th locker, and so on until all 1,000 students passed through the school.

Which lockers were open and why?

[Placebo]

1. Spork.

:crazy:

[gmoney]

2. They're all closed!

Because April 1 was a Sunday and school was closed!

[DustMolecule]

4. Hahaha.....

nah, that's not it....silly :-P

[Massacure]

3. It would go something like:

1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5
c c c c c c c c c c c c c c c c c c c c c c c c c
o o o o o o o o o o o o o o o o o o o o o o o o o
o c c c o o o c c c o o o c c c o o o c c c o o c
o c c o o o o o c c o c o c c o o o o o c c o c o

The last line is the order for the first 25 lockers.

[DustMolecule]

5. nope.....sorry, that's not it

.

[Massacure]

6. All of them were opened!

[Commie Pinko Dirtbag]

7. All closed but the 1st (nt)

[DustMolecule]

8. No guessing......you've gotta explain the 'why' with the answer

.

[Kadie]

9. All of them are open because

they were cheap lockers, and after being opened and shut so many times all the doors fell off.

[Peanut Gallery]

18. LOL

n/t

[Kadie]

10. Oh wait, they are all closed

because the 1st student would have only been able to open one locker, his own, the only one he had a combination to. One of the following students would have closed it. So, they are all closed.

[DustMolecule]

11. *sigh* Did I forget to mention this is UTOPIA High School.....

...no one steals in Utopia, so they don't have locks.

[Kadie]

12. Darn. Thought I had it. hmmmmmm

[Commie Pinko Dirtbag]

13. I smell prime numbers

Prime numbered lockers will be closed, because none of the previous numbered students would be a divisor of the locker number. They'd be opened by student 1 and closed by student p and that's all.

About the others... it depends whether they have an odd or even number of divisors. 4 has an odd number of divisors (1, 2, 4) so it will be open. 8 has an even number of divisors (1, 2, 4, 8) so it will be closed.

But how many numbers are like that? Argh, I hit bedrock.

[DustMolecule]

14. Your definitely on the right track!

someone let me know when y'all want the answer, in case you don't want to figure it out anymore).

[Commie Pinko Dirtbag]

16. Uncle (nt)

[pfitz59]

15. Prime number lockers would be closed....

All others open.. (Don't ask me to list the prime numbers to 1000)

[Commie Pinko Dirtbag]

17. No. Look at locker #8 for instance.

It will be opened by #1, closed by #2, opened by #4, and closed by #8.

[DustMolecule]

19. The Answer is:

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

[pres2032]

20. thanks, my head hurts now

Hehe, i never liked math

[Pithlet]

21. They're all square numbers?

This was a good one.

[BrewerJohn]

22. The key seems to be that (more precisely stated)

the lockers that are open at the end have an odd number of distinct divisors. Numbers that are squares have a repeated divisor, which "collapses" into one when counting the number of distinct divisors. The interesting point of the puzzle is that it must be that only in the case of a square is the number of distinct divisors odd (in this case, the number is 3). So there are no numbers that have five distinct divisors, or seven? Or are the only ones all greater than 1000?

(incurable math geek here, in case it's not obvious)

[Commie Pinko Dirtbag]

23. OK, every number x is equal to p1^n1 x p2^n2 x ...

The distinct divisors are made like this:

- We choose a number between 0 and n1, inclusive, for p1. Because p1^(anything between 0 and n1) will divide x.
- Same for n2, n3 etc.
- Therefore, the number of distinct divisors is (n1+1)(n2+1)(n3+1)...
- Which will only be odd if all factors are odd. That means all of n1, n2, ... must be EVEN.
- Number with all prime exponents even = square number.

Therefore the answer is that all perfect square numbered lockers are open. The number of open lockers is the SQUARE ROOT of the number of lockers.

My, what difference a good night's sleep makes.