100 doors in a row Interview puzzle

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You have 100 doors in a row that are all initially closed. You make 100
passes by the doors starting with the first door every time. the first time through
you visit every door and toggle the door (if the door is closed, you open it, if its
open, you close it). the second time you only visit every 2nd door (door #2, #4,
#6). the third time, every 3rd door (door #3, #6, #9), etc, until you only visit the
100th door.
Question: what state are the doors in after the last pass? which are open which
are closed?

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  • 1 Answer(s)

    You can figure out that for any given door, say door #42, you will visit it
    for every divisor it has. so 42 has 1 & 42, 2 & 21, 3 & 14, 6 &7. so on pass 1 i will
    open the door, pass 2 i will close it, pass 3 open, pass 6 close, pass 7 open,
    pass 14 close, pass 21 open, pass 42 close. for every pair of divisors the door
    will just end up back in its initial state. so you might think that every door will end
    up closed? well what about door #9. 9 has the divisors 1 & 9, 3& 3. but 3 is
    repeated because 9 is a perfect square, so you will only visit door #9, on pass 1,
    3, and 9… leaving it open at the end. only perfect square doors will be open at
    the end.

    pnikam Expert Answered on 22nd August 2015.
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