imagine you are at a school that still has student lockers. there are 1000 lockers, all shut and unlocked and?

Question

1000 students.
1. Suppose the first student goes along and opens evvery locker.
2. the second studentv goes along and shuts every other locker beginning with number 2.
3. the third student then goes along and changes the state of every third locker beginning with the number 3. (if the locker is open, the student shuts it and if it's closed the student will open it)
4. the fourth student changes the state of every fourth locker beginning with number 4.

if this continues until all 1000 students have followed the pattern with these lockers, which lockers will be open and which will be shut at the end? why?

Answer 1

In this case, after all 1,000 students are done, every locker will be closed if it has an even number of factors (e.g., 1, 2, 5, 10 for locker 10 - four state-changes leaves it closed), and open if it has an odd number of factors.

Only lockers whose number is a perfect square have an odd number of factors (e.g., 1, 3, 9 for locker 9).

Answer 2

all numbers that are perfect squares between 1 and 1000 will be open. only perfect squares have an odd number of factors and since all doors originally started in the closed position only an odd number of state changes will leave them in the open position.

Answer 3

Try it just for, say, the first 1 lockers.

You will see a pattern.

1, 4, 9,....

See the pattern?.


OOOOPSS, I misread the question. Your question is very close to a famous problem, but not quite as you stated. Sorry....too much haste....

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