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To start off with: A Classic

 
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Michael Chu



Joined: 10 May 2005
Posts: 1617
Location: Austin, TX (USA)

PostPosted: Thu May 12, 2005 8:31 am    Post subject: To start off with: A Classic Reply with quote

You are in a corridor with 100 closed doors. You are instructed to open every single door and then return to where you started. You are then instructed to close every second door and then return to where you started. Then you are instructed to go to every third door and either open or close it depending on if it is closed or open. This continues with every fourth door, fifth door, and so one until you are finally instructed either open or close the one hundredth door (depending on if it is closed or open at this point). Assuming you followed all the instructions properly, how many lockers are now open?
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capstinence



Joined: 12 May 2005
Posts: 44
Location: Los Angeles, CA

PostPosted: Thu May 12, 2005 8:45 pm    Post subject: Reply with quote

My first thought is to say no lockers are open, since we are opening doors.

But I've been trying to work it out on paper and find some sort of pattern. I started with 10 doors, then did 20.. and the percentage of doors open changed slightly, so I can't just do that. Also, more than one door will change only until you hit the halfway mark (ie. there's only one 51st door, 52nd door, and so on), so that must be helpful somehow.

I'll have the kids I tutor do the tedious work. Wink
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capstinence



Joined: 12 May 2005
Posts: 44
Location: Los Angeles, CA

PostPosted: Fri May 13, 2005 5:46 am    Post subject: Reply with quote

Got it, 10.

But there has to be an easier way than the way I did it.
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Michael Chu



Joined: 10 May 2005
Posts: 1617
Location: Austin, TX (USA)

PostPosted: Fri May 13, 2005 6:50 pm    Post subject: Reply with quote

10 is correct. And there probably is a faster way to do it.

I assume you actually did the exercise? So, perhaps 100 doors isn't enough - how about 10000 doors? :sly:

So, a hint: take a look at which doors remain open. Which doors were they?
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capstinence



Joined: 12 May 2005
Posts: 44
Location: Los Angeles, CA

PostPosted: Fri May 13, 2005 8:04 pm    Post subject: Reply with quote

Oh wow... I'm an idiot.

Any number that you can take the square root of and be given a whole number.. a perfect square. *smacks forehead* I got so caught up in the exercise that I didn't even pay attention to which doors they were.
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Michael Chu



Joined: 10 May 2005
Posts: 1617
Location: Austin, TX (USA)

PostPosted: Fri May 13, 2005 8:20 pm    Post subject: Reply with quote

Yep, and since there are sqrt(n) squares between 1 and n, there are sqrt(n) open doors.
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capstinence



Joined: 12 May 2005
Posts: 44
Location: Los Angeles, CA

PostPosted: Sat May 14, 2005 5:23 am    Post subject: Reply with quote

Man, that is tricky.
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