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Math Brain Teaser

  1. Nov 23, 2003 #1
    1. A new high school has just been completed. There are 1,000 lockers in the long hall of the school and they have been numbered from 1 to 1,000. During lunch, the 1,000 students decide to try an experiment.

    - The first student, student 1, runs down the row of lockers and opens every door.
    - Student 2 closes the doors of lockers 2, 4, 6, 8 and so on to the end of the line.
    - Student 3 changes the state of the doors of lockers 3, 6, 9, 12 and so on to the end of the line. (the student opens the door if it is closed and closes the door if it is opened)
    - Studnet 4 changes the state of the doors 4, 8, 12, 16 and so on. Student 5 changes the state of every fifth door, student 6 changes the state of every sixth, and so on until all 1000 students have had a turn.

    When the students are finished, which lockers doors are open?
    I was REALLY tempted to sit tehr are write out numbers 1-1000 but decided there was an easier way...
  2. jcsd
  3. Nov 24, 2003 #2
    Try the case when there are only 10 lockers and try to find out a pattern. Then try to explain why the pattern is like this. :smile:
  4. Nov 24, 2003 #3


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    There won't be a pattern. Take the factors of each number, include 1 and the number itself. Count the number of factors. This is the number of "state" flips. If it is even, the locker is closed. If it is odd, the locker is open.

  5. Nov 24, 2003 #4


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    There most certainly is a pattern.
  6. Nov 25, 2003 #5


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    I had to do a few more lockers. You're right Hurkyl. For anyone else trying this, more power two ya'. :wink:

  7. Nov 26, 2003 #6
    One certainty

    Prime numbered lockers will be closed.
  8. Nov 26, 2003 #7


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    There's more two it than that. Many other lockers will be closed too.

  9. Jun 1, 2010 #8
    All the square numbers will be open............
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