Count Doors

Replies

• 

  tech_vaibhav_ee

  31 doors...
  maths behind it:no. of perfect squares b\w 1 and thousand...
  logic: the no. of factors of a no. is always a even no. unless it is a perfect square..and the door is open by those people whose no. are factor of the door to be opened...
• 

  pradeep_agrawal

  tech_vaibhav_ee, you are really good at puzzle solving.
  
  Both the answer and the explanation given above is correct.
  
  -Pradeep
• 

  shalini_goel14

  Sorry, I couldn't get the explanation, can anyone explain it CLEARLY again ?
• 

  vishnu priya

  I also join with shalini.I too did not understand it clearly.
• 

  pradeep_agrawal

  Let me explain the logic behind the solution.
  
  A person can visit a door only if the door number is completely divisible by the number associated with the person. In other word we can say that, a door can be visited by only by persons whose number completely divides the door number.
  
  The number of person visiting any door will be equal to the number of factors of that door number, e.g., consider door number 10, there are four factors of number 10 (i.e., 1, 2, 5, and 10) and thus the door number 10 can be visited by four person with number 1, 2, 5, and 10.
  
  Lets consider opening a door or closing a door as one operation.
  
  As initially all the doors are close, so a door will be in open state at the end if it has been operated odd number of times (1, 3, 5, ...), i.e., odd number of people have visited the door.
  
  Now odd number of people will visit a door if door has odd number of factors.
  
  If we do some mathematics we will find that only the numbers which are perfect square of some other number have odd number of factors (e.g., 25 has 3 factors as 1, 5, and 25) and all other numbers have even number of factors.
  
  So only those doors will be in open state whose number is a perfect square of some other number. Now there are only 31 such numbers between 1 to 1000 (1, 4, 9, 16, 25 ... 900, 961) so 31 doors will be in open state.
  
  -Pradeep
• 

  tech_vaibhav_ee

  i didnt gave the solution properly and just some hints so that others can also try the puzzle...
  pradeep i think u should also wait 4 atleast a week before giving the whole solution as it gives more time 4 people to solve the puzzle...
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  shalini_goel14

  @Pradeep Sir : Cool, what an explanation. Thanks , now it is cleared to me. 😀