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From: tallman@dspo.UUCP
Newsgroups: net.puzzle
Subject: Re: 1000 Lockers (SPOILER)
Message-ID: <215@dspo.UUCP>
Date: Tue, 23-Apr-85 13:18:06 EST
Article-I.D.: dspo.215
Posted: Tue Apr 23 13:18:06 1985
Date-Received: Fri, 26-Apr-85 03:28:58 EST
References: <233@bbnccv.UUCP>
Organization: Los Alamos National Laboratory
Lines: 22

> Lining the corridor of a school are 1000 lockers.  The first student
> to arrive at school one morning decides to open the doors to all 1000
> lockers.  The second student to arrive that morning decides to go down
> the hall and close the door of every second locker...
> As the students arrive, this process continues...until the 1000th
> student... Now, after all this has been done, which lockers are open,
> which are closed, and why?

All lockers whose numbers are perfect squares will be open (1,4,9...)
and all others will be closed.

The reasoning is as follows - 
A locker will be closed if it has an even number of divisors including
itself and 1.  It will be open if it has an odd number.  Suppose a
number n has a divisor k.  Then it also has a divisor n/k. So all
divisors of n pair up, unless there exists k = n/k, i.e. n is the
perfect square k*k.

-- 
C. David Tallman - dspo!tallman@LANL  or {ucbvax!unmvax,ihnp4}!lanl!dspo!tallman
Los Alamos National Laboratory - E-10/Data Systems
Los Alamos, New Mexico  -  (505) 667-8495