Word problem

Question

Word problem

A high school has 4657 students and 4657 lockers, all of which are closed. The students line up and walk one at a time down the hall that contains the lockers. The first student opens all the lockers. The second student closes every other locker starting with the second one. The third student changes (closes if opened, opens of closed) every third locker starting with the third. The fourth student changes every fourth locker starting with the fourth. This pattern continues until all 4657 students have passed through the hallway.

In the end, how many lockers are open?
1

Answer 1

First of all, look at my comment. I'm going to assume my reasoning is correct unless someone shows me a convincing argument against my reasoning.

 

Take the first 16 lockers and the first 16 students.

A    B    C    D    E    F    G    H    I    J    K    L    M    N    O    P

o    o     o    o    o    o     o    o    o   o    o    o     o    o     o   o    1st

o    c     o     c    o    c     o    c    o   c    o     c     o    c     o   c    2nd

o    c     c     c    o    o     o    c    c   c    o     o     o    c     c   c    3rd

o    c     c     o    o    o     o    o    c   c    o     c     o    c     c   o    4th

o    c     c     o    c    o     o     o   c   o    o     c     o    c     o   o    5th

o    c     c     o    c    c     o     o   c   o    o     o     o     c    o   o    6th

o    c     c     o    c    c     c     o   c   o    o     o     o     o    o   o    7th

o    c     c     o    c    c     c     c   c   o    o     o     o     o    o   c    8th

o    c     c     o    c    c     c     c   o   o    o     o     o     o    o   c    9th

o    c     c     o    c    c     c     c   o   c    o     o      o     o    o   c    10th

o    c     c     o    c    c     c     c   o   c    c     o      o     o    o   c     11th

o    c     c     o    c    c     c     c   o   c    c     c      o      o    o   c     12th

o    c     c     o    c    c     c     c   o   c    c     c      c      o    o   c      13th

o    c     c     o    c    c     c     c   o   c    c     c      c      c     o   c      14th

o    c     c     o    c    c     c     c   o   c    c     c      c      c     c   c      15th

o    c     c     o    c    c     c     c   o   c    c     c      c      c     c   o      16th 

I notice a pattern here.

The 1st, 4th, 9th, and 16th are all open.

1,4,9,and 16 are perfect squares. This makes me believe every locker whose number is a perfect square will be open.

The following lockers will be open:

#'s 1,4,9,16,25,36,49,64,81,100,121,144,169,196,225,256,289,324,361,400,441,484,529,576,625,676,729,784

841,900(30²)...4356(66²),4489(67²),4624(68²)(69² exceeds 4657).

There will be 68 lockers open.

 

 

Answer 2

As I see the problem, the only locker that is left open is the first and nobody touches it so I would answer that only one locker is left open.