1000 LockersDate: 11/06/97 at 23:35:30 From: Kate Thorsheim Subject: Word problem There are 1000 lockers in a high school with 1000 students. The problem begins with the first student opening all 1000 lockers; next the second student closes lockers 2,4,6,8,10 and so on to locker 1000; the third student changes the state (opens lockers closed, closes lockers open) on lockers 3,6,9,12,15 and so on; the fourth student changes the state of lockers 4,8,12,16 and so on. This goes on until every student has had a turn. How many lockers will be open at the end? What is the formula? I can't figure out the pattern. Kate Date: 11/07/97 at 13:41:12 From: Doctor Bruce Subject: Re: Word problem Hello Kate, I enjoyed thinking about this problem when I first heard it some years ago. One thing we can 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!). Good luck! -Doctor Bruce, The Math Forum Check out our web site! http://mathforum.org/dr.math/ Date: 11/07/97 at 19:46:25 From: Anonymous Subject: Re: Word problem Dr. Math, Thank you for your prompt response. Kate |
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