Another Math Problem

[T. R. Oglodyte Moderator Find all posts View Profile Close T. R. Oglodyte]

OK - here's a harder one.

At Lincoln High School there are 100 lockers in the hallway in the Social Studies wing. At the end of the school year, the lockers are all unlocked, the students having cleaned them out. The next day starts summer vacation, which lasts for 100 days.

On the first day of school vacation, a janitor locks every locker.

On the second day of school vacation, the janitor unlocks every second locker, starting with locker #2.

On the third day of school vacation, the janitor checks every third locker, starting with locker #3. If the locker is locked, he unlocks it. If it is unlocked, he locks it.

On the fourth day of school vacation, the janitor checks every fourth locker, starting with locker #4, locking it if unlocked, unlocking it if locked.

Throughout the remainder of the school vacation he continues this process - on day n he checks every nth locker, starting with locker #n - locking if unlocked and locking if unlocked.

At the end of the last day of vacation (i.e., at the end of day 100, after the janitor has completed his 100th time of checking lockers), how many lockers are locked and how many are unlocked.

Figure out the answer deductively - not by working it out on paper.

[jrdc Junior Member Find all posts View Profile Close jrdc]

First of all, how come I never found the school with 100 days of summer vacation?

They're all unlocked by the end of vacation...just in time for everyone to get back to school to cram them full of junk & half eaten PB & J's.

[bigfrank Administrator Find all posts View Profile Close bigfrank]

Steve I moved your post to a new thread so people will see it.

[Jya-Ning Silver Contributor Find all posts View Profile Close Jya-Ning]

O.K. first try

Look like if it is prime number(?, hopefully it is correct term), it will be done exactly once,
so 1, 2, 3, 5, 7, 11, 13 ... are prime number, all except #1 is unlocked

The rest will depend on how many prime number is on them, if it is even it will be locked, if it is odd, it will be unlocked

so 4 has 2 * 2 or 2 prime number, so 4 is locked
100 is 2 * 2 * 5 * 5 or 4 prime number so 100 is locked.
12 is 2 * 2 * 3 or 3 prime number, so it will be unlocked?

Too hard for me to guess how many are locked and how many are not based on this without using pen or spreadsheet.

Will need to think again.

On the other hand, if I was in that school, I will come at the last day and lock them up for fun, so all locked

Jya-Ning

[T. R. Oglodyte Moderator Find all posts View Profile Close T. R. Oglodyte]

O.K. first try

Look like if it is prime number(?, hopefully it is correct term), it will be done exactly once,
so 1, 2, 3, 5, 7, 11, 13 ... are prime number, all except #1 is unlocked

The rest will depend on how many prime number is on them, if it is even it will be locked, if it is odd, it will be unlocked

so 4 has 2 * 2 or 2 prime number, so 4 is locked
100 is 2 * 2 * 5 * 5 or 4 prime number so 100 is locked.
12 is 2 * 2 * 3 or 3 prime number, so it will be unlocked?

Too hard for me to guess how many are locked and how many are not based on this without using pen or spreadsheet.

Will need to think again.

On the other hand, if I was in that school, I will come at the last day and lock them up for fun, so all locked

Jya-Ning

You don't need pen or spreadsheet. There's a straightforward solution that can be worked out in your head.

Hint: it's not a prime numbers problem.

[Jya-Ning Silver Contributor Find all posts View Profile Close Jya-Ning]

Ah, I have my drink.

So 2, 3 is unlocked, the rest are locked.

Jya-Ning

[T. R. Oglodyte Moderator Find all posts View Profile Close T. R. Oglodyte]

Ah, I have my drink.

So 2, 3 is unlocked, the rest are locked.

Jya-Ning

Nope.

[BILL_B Member Find all posts View Profile Close BILL_B]

Steve,

I agree with Jya-ning.

After day 4 locker#'s 1 and 4 are locked and locker#'s 2 and 3 are unlocked. On subsequent days each successive locker is locked if it's unlocked based on the rule you stated.


Throughout the remainder of the school vacation he continues this process - on day n he checks every nth locker, starting with locker #n - locking if unlocked and locking if unlocked

Bill

[T. R. Oglodyte Moderator Find all posts View Profile Close T. R. Oglodyte]

Steve,

I agree with Jya-ning.

After day 4 locker#'s 1 and 4 are locked and locker#'s 2 and 3 are unlocked. On subsequent days each successive locker is locked if it's unlocked based on the rule you stated.


Throughout the remainder of the school vacation he continues this process - on day n he checks every nth locker, starting with locker #n - locking if unlocked and locking if unlocked

Bill

Ga-dung!!!!!! There's a reason why I use the avatar I've selected.

I'm a terrible proof-reader. My lysdexia afflicts me in my old age. Did I ever mention that last Christmas I celebrated the birth of the son of Dog??

Getting back to the riddle. The janitor locks if unlocked; unlocks if locked. Sorry!!!

[Elan Senior Member Find all posts View Profile Close Elan]

The lockers that ultimately end up locked will have needed to have been touched an even number of times (unlocking, then re-locking). The only numbers that have an even number of factors (not counting 1, since the lockers were originally locked on day 1), are the perfect squares. Their factors are themselves and their square roots (and other perfect squares). So I would guess that there are 10 locked lockers -- 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100. Or maybe not..........?????

[Jya-Ning Silver Contributor Find all posts View Profile Close Jya-Ning]

Jya-Ning

[T. R. Oglodyte Moderator Find all posts View Profile Close T. R. Oglodyte]

The lockers that ultimately end up locked will have needed to have been touched an even number of times (unlocking, then re-locking). The only numbers that have an even number of factors (not counting 1, since the lockers were originally locked on day 1), are the perfect squares. Their factors are themselves and their square roots (and other perfect squares). So I would guess that there are 10 locked lockers -- 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100. Or maybe not..........?????

Correct. Both the number and the reasoning.

• After a janitor visits a locker for the nth time, the locker will be unlocked if n is even and will be locked if n is odd.
  
• The total number of times a janitor visits a locker is the same as the number of factors associated with the locker number.
  
• Since all numbers except for perfect squares have an even number of factors, the only lockers the janitor checks an odd number of times are the lockers for which the locker number is a perfect square.

[Elan Senior Member Find all posts View Profile Close Elan]

All I can say is it's a good thing my reading comprehension isn't any better than it is, or I would've seen what Jya-Ning saw. Another sign of deteriorating mental accuity......

[BILL_B Member Find all posts View Profile Close BILL_B]

I liked it better when I thought it was a trick question

Now that it's been explained I realize that I wouldn't have figured it out in a million years

[jrdc Junior Member Find all posts View Profile Close jrdc]

I liked it better when I thought it was a trick question

Now that it's been explained I realize that I wouldn't have figured it out in a million years

LOL! I'm with you...I was thinking what Jya-Ning initially thought & still got it wrong!