Opened And Closed Lockers
A school hallway has a thousand lockers. A student goes down the hallway opening every locker. Then a second student goes down the hallway closing
every other locker. A third student goes down the hallway, and at every third locker, he opens it if it was closed and closes it if it was opened. This continues on
for a total of 1000 students opening and closing lockers. How many lockers are opened when the students are finished?
Solution
The lockers which are opened are the ones which have been "altered" an odd number of times. This means that their locker numbers have an odd number of divisors, which
means they are all perfect squares. There are 31 perfect squares less than 1000.
Isn't that slick?
"Slick Math" is written by Douglas Twitchell, and hosted at The Problem Site.
Contents copyright 2008 by Douglas Twitchell. Contents of this page may not be reproduced without permission of the author.
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