# Opening and closing doors

1. Dec 3, 2008

### maze

Hey i posted this puzzle a few days ago in another forum, but the stickied thread is completely dead and I don't think anyone even reads it anymore, so I'm going to post it here. Plus I think this is the more proper forum as it is more of a puzzle than a math problem (though it is both).

PUZZLE:
In a very long hallway, there are 1000 doors all initially open.
First, you close every door.
Second, you open every other door.
Next, you toggle the state of every 3rd door (open it if it is closed and close it if it is open),
Next, you toggle every 4th door,
and you continue this process, at the toggling the state of every nth door at the n'th step.

At the end of this process (when n=1000), how many doors are open?

Here is a diagram:

2. Dec 3, 2008

### CompuChip

I think I found the result numerically...

969 are open

is that right? Now let's see if I can derive it without computer

Cool, I didn't know spoiler tags worked here![/edit]

3. Dec 3, 2008

### maze

Yep numerically that's right. Now what if there are 10^100000 doors? ;)

4. Dec 3, 2008

### davee123

For N doors, the number of open doors is: N-FLOOR(SQRT(N))

DaveE

5. Dec 8, 2008

### maze

correct indeed! nice one