Question 51262: In a particular school there are exactly 1000 lockers numbered 1 to 1000. In this school there happen to be exactly 1000 students. One day the math teacher has all of the students line up in the hallway. His instructions are as follows:
"The locker doors are all closed at this time. I would like the first student to go to every locker and open the door. Next, I would like the second student to go to every second locker (#2,#4,#6...)and close the door. Then I would like the third student to go to every third Locker (#3,#6,#9...)and change the current status of the locker door ( if the door is open, close it, if it's closed then open it. Likewise, the fourth student will go to every fourth locker and change the door's status. this pattern will continue until all 1000 students have changed the door status of their appropriate lockers."
after all the students have gone through this process, which of the 1000 lockers will have open doors and which will have closed doors?
The only logical answer I could come up with is locker door #1 will be open and doors #2-1000 will be closed. Is this right?
Thanks so much for any input you can offer.
Bridget
Found 2 solutions by rapaljer, richard1234:
Answer by rapaljer(4671)   (Show Source):
You can put this solution on YOUR website! This also is a CLASSIC problem!! To do this one, you should write the numbers from 1 to 25 vertically, representing each locker. Leave space at the top of this and write the numbers of each student who is going to open and close the doors, from 1 to 25 (maybe less, since you will probably figure out the pattern way before you get to 25!).
Now, in column 1, label each door with the letter "O" for open. In column 2, label each of the even doors with the letter "C" for closed. In column 3, label the multiples of 3 with the opposite of the previous letter for the status of the door. Continue until you see a pattern emerge. I hate to give you the answer before you have a chance to do this exercise, since it is a truly GREAT discovery problem!!
You are correct that most of the doors will end up closed, but there ARE indeed many doors that will be left open, and it is a GREAT pattern!! If you try this, and want the answer, send me a note, and I'll reply by Email.
R^2 at SCC
Answer by richard1234(7193)  (Show Source):
You can put this solution on YOUR website! I believe I saw this problem on a previous AIME exam (American Invitational Mathematics Examination). This is indeed a classic problem. The solution involves how many integer factors each number has, but I'll leave up the rest to you (Try small cases!).
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