There is a school with 1,000 students and 1,000 lockers. On
the first day of term the headteacher asks the first student to go along and open every single locker, he asks the second to
go to every second locker and close it, the third to go to every
third locker and close it if it is open or open it if it is closed,
the fourth to go to the fourth locker and so on. The process is
completed with the thousandth student. How many lockers
are open at the end?
Answers
Answer:
1000
Step-by-step explanation:
because there are 1000 lockers for 1000 students
Answer:
Step-by-step explanation:
Student #1: Go to every locker and open it.
Student #2: Go to every second locker and close it.
Student #3: Go to every third locker. If it is closed, open it, and if it is open, close it.
Student #4: Go to every fourth locker. If it is closed, open it, and if it is open, close it.
This goes on until Student #1000 is finished.
After the closing ceremonies are finished, the principal walks through the school and closes each locker that is left open. How many lockers will the principal close?
Hint:
Test it out with only ten lockers and ten students and see if you can find a pattern.
Solution:
The principal only needs to close the lockers whose numbers are perfect squares. This means the solution is as easy as finding the square root of the highest possible perfect square within 1000.
31*31 = 961
32*32 = 1024
Therefore, 31 is the number of lockers the principal has to close.
Shown below is the solution with ten lockers and students.
X=Closed locker
O=Open locker
Here is what happens to the lockers from the first to the third student…
see the screen shot
The lockers left open after the tenth student is finished are lockers 1,4, and 9 — the only three perfect squares below 10.
The solution might become more obvious if you look at a specific locker and add together the number of students who have opened it and the number of students who have closed it.
The sum is equal to how many factors the number of that locker has. Each time a student opens or closes a certain locker, it is implying that the number assigned to the student is a factor of that locker number.
Since the lockers are closed to begin with, any time a locker number has an even number of factors, it will end up closed. Numbers with an odd number of factors will end up open. All perfect squares have an odd number of factors, which is why the lockers with these numbers end up open, while others end up closed.
