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### Hard Maths Puzzle

Hard Maths Puzzle - 3 October

A high school has a strange principal. On the first day, he has his students perform an odd opening day ceremony:

There are one thousand lockers and one thousand students in the school. The principal asks the first student to go to every locker and open it. Then he has the second student go to every second locker and close it. The third goes to every third locker and, if it is closed, he opens it, and if it is open, he closes it. The fourth student does this to every fourth locker, and so on. After the process is completed with the thousandth student, how many lockers are open?

1. all numbers with odd no.of factors will be open

so all perfect squares less than 1000 will be open.So no. of open lockers =31 since 31^2=961 the highest perfect square less than 1000

1. Because all primes have even number of factors so do all other numbers except ones that are perfect squares. good answer.

2. Correct all perfect squares from 1 to 1000. i.e 1, 4, 9, 16,............961. total 31 lockers

2. Hardest Maths Puzzle

3. 31 locks will be opened....

4. 31 lockers are open, and their location will be 1, 4, 9,25, 36,47,64,81..........till 961

1. And 16,49. Not 47, that ones closed. Mathswiz is correct...

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8. All perfect SQUARE (31)

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10. 16 will be found open or close , I think it is open if it close than please explain...

11. the actuall ans pysically is none of these is open nor closed

12. How to solve it?

13. instead of this, very first student should lock principal mouth...

14. But why will it be the perfect squares....how can this be actually mathematically proved?

15. Each locker will remain open if it has an odd number of "pupil visits" (number of times a pupil visits a locker and opens or closes it):

One visit: locker open
Two visits: locker closed
Three: open
Four: closed
Etc.

Each locker L has n number of pupil visits where n is the number of factors of L:

Locker 18 has 6 visits, by pupils 1, 2, 3, 6, 9, 18.
Locker 31 has 2 visits, by pupils 1, 31.

If n is even, the locked remains closed.

For any given number x we can group its factors in pairs (d, x/d) meaning that the number of factors is even, unless x=y*y when y pairs with itself. This gives x an odd number of factors.

If L has an odd number of factors then the locker remains open. Only when L is a square number does it have an odd number of pupil visits. Therefore the only lockers that remain open are those that are square numbers 1, 4, 9, 16, 25... etc.