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Math Help - Lockers

  1. #1
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    Lockers

    A school has 1000 students and 1000 student lockers. The lockers are in
    a line in a long corridor and are numbered from 1 to 1000.

    Initially all the lockers are closed (but unlocked).
    The first student walks along the corridor and opens every locker.

    The second student then walks along the corridor and closes every second locker, i.e. closes lockers 2, 4, 6, etc. At that point there are 500 lockers that are open and 500 that are closed.

    The third student then walks along the corridor, changing the state of every third locker.Thus s/he closes locker 3 (which had been left open by the _rst student), opens locker 6 (closed by the second student), closes locker 9, etc.

    All the remaining students now walk by in order, with the kth student changing the state of every kth locker, and this continues until all 1000 students have walked along the corridor.

    i) How many lockers are closed immediately after the third student has walked along the corridor? Explain your reasoning.

    (ii) How many lockers are closed immediately after the fourth student has walked along the corridor? Explain your reasoning.

    (iii) At the end (after all 1000 students have passed), what is the state of locker 100? Explain your reasoning.

    (iv) After the hundredth student has walked along the corridor, what is the state of locker 1000? Explain your reasoning.
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  2. #2
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    I am not sure about my answers, this is what I have:


    i) student 3 opens all even multiples of 3, closes odd multiples. He changes 333 lockers, and 333/3 = odd, so last thing he does is close a locker, which isnt cancelled by a further opening.


    so 1 more closed at the end. ==> 501 closed

    ii) All multiples of 3 and 4 will be closed, and multiples of 4 not multiples of 3 opened

    all numbers of the form 12a are multiples of 3 and 4

    1000 div by 4 250 times, so 250 lockers changed


    1000 is div by 12, 83 times, so 83 closed

    250 - 83 = 167 opened

    so the total is: 167 - 83 = 83 lockers opened which not cancelled by another closing

    so 501 - 84 = 417 lockers closed at the end

    iii) 1000 is div by: 1, 2, 4, 5, 10, 20, 25, 50, 100

    changed 9 times, open, close ... open, close, 100th student opens

    so it is open

    iv) 1000 div by: 1. 2. 4. 5. 8, 10, 20, 25, 40, 50, 100

    so changed 11 times

    opened by 100th student, so it is open

    Any help please whether on these are right? Also, could someone present with a good method to solve this question? Thank you
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  3. #3
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    Quote Originally Posted by Aquafina View Post
    A school has 1000 students and 1000 student lockers. The lockers are in
    a line in a long corridor and are numbered from 1 to 1000.

    Initially all the lockers are closed (but unlocked).
    The first student walks along the corridor and opens every locker.

    The second student then walks along the corridor and closes every second locker, i.e. closes lockers 2, 4, 6, etc. At that point there are 500 lockers that are open and 500 that are closed.

    The third student then walks along the corridor, changing the state of every third locker.Thus s/he closes locker 3 (which had been left open by the _rst student), opens locker 6 (closed by the second student), closes locker 9, etc.

    All the remaining students now walk by in order, with the kth student changing the state of every kth locker, and this continues until all 1000 students have walked along the corridor.

    i) How many lockers are closed immediately after the third student has walked along the corridor? Explain your reasoning.

    (ii) How many lockers are closed immediately after the fourth student has walked along the corridor? Explain your reasoning.

    (iii) At the end (after all 1000 students have passed), what is the state of locker 100? Explain your reasoning.

    (iv) After the hundredth student has walked along the corridor, what is the state of locker 1000? Explain your reasoning.
    Discussed at MHF a number of times. See here http://www.mathhelpforum.com/math-he...r-problem.html
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