Wednesday

February 10, 2016
Posted by **Ibby** on Thursday, October 25, 2007 at 10:04am.

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**josh**, Thursday, October 25, 2007 at 10:17amummmm all 1,000 cuz if each opens and closes all 1,000 should be opened

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**Ibby**, Thursday, October 25, 2007 at 11:27amNo, cuz if the second student goes in and shuts every other locker then the second locker would be shut at the end

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**Reiny**, Thursday, October 25, 2007 at 11:53amOnly lockers whose numbers are perfect squares will be open

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**Ibby**, Thursday, October 25, 2007 at 12:54pmThank You Reiny

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**tchrwill**, Thursday, October 25, 2007 at 2:46pm<< There are 1000 lockers in the long hall of Westfalls High. In preparation for the beginning of school, the janitor cleans the lockers and paints fresh numbers on the locker doors. The lockers are numbered from 1 to 1000. When the Westfalls High students arrice from summer vacation, they decide to celebrate the beginning of school by working off some energy. The first student 1, runs down the row of lockers and opens every door. Student 2 closes the door of locker 2 and every other locker after that, 2, 4, 6, 8, and so on to the end of the line. Student 3 opens the door of locker 3 and every third locker after that, 3, 9, 12, 15, and so on the end of the line. (The student opens the door if it is closed and closes the door if it is open.?)

Student 4 closes the door of locker 4 and every fourth locker after that, 4, 8, 12, 16, and so on.

Student 5 opens the door of locker 5 and every fifth locker after that, student 6 changes the state of every sixth door starting with locker 6, and so on until all 1000 students have had a turn.

When the students are finished, which locker doors are open? What is the pattern?

We could take the long route and discuss the evolution of the answer but lets try a shorter path.

Lets look at locker number "n". All the lockers are locked to begin with. Since all the lockers are opened on the 1st pass, locker "n" is now open. For locker "n" to be closed on the 2nd pass, n must be divisible by 2. For the locker to be opened on the 3rd pass, it must be divisible by 3. For the locker to be closed on the 4th pass, it must be divisible by 4. Clearly, the locker is either opened or closed as long as the locker number is divisible by each successive divisor of "n". After the one thousandth student has made his contribution to the celebration, locker "n" will only be open if it was acted upon an odd number of times. We can therefore conclude that locker "n" will be open if, and only if, the number "n" has an odd number of factors or divisors. But the only numbers that have an odd number of factors/divisors are the perfect squares. Thus, if "n" is open, it is one of the perfect squares. The lockers that remain open are therefore numbers 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, on up to 961.

For each locker number, find all of the exact divisors including 1 and the number itself. If the number of divisors is odd, then the number of people who reversed the locker is odd, and the locker is open. If the number of divisors is even, then the number of people who reversed the locker is even, and it is closed. The only numbers with an odd number of divisors are the perfect squares. Therefore, the lockers that remain open are those identified by the perfect squares.

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**Anonymous**, Thursday, February 19, 2015 at 8:46pmA very long hallway is lined with 10,000 lockers which are all closed. The first person comes along and opens all the lockers. The second person shuts the even lockers (2nd, 4th, 6th, 8th … lockers) and leaves the rest open. The third person changes the lockers which are multiples of 3. (3rd, 6th, 9th, 12th… lockers) If one of these lockers was open he closed it, and if it was closed he opened it. The 4th person changes the lockers which are multiples of 4. (4th, 8th, 12th, 16th… lockers) The fifth person changes the lockers which are multiples of 5. (5th, 10th, 15th, 20th … lockers) This continues with people opening and closing lockers. Which lockers remain open after the 10,000th person goes down the hallway?