This is a classic logic problem for you to think about. You may find it helpful to represent the various scenarios in a visual way.

• A school has 1000 students and 1000 closed lockers numbered 1 to 1000.
• The first student walks along the corridor and opens every locker.
• The second student walks along and closes every second locker.
• The third student walks along and changes the state of every third locker (ie. an open locker is closed and a closed locker is opened) .
• Remaining students walk along, and the kth student change every kth locker, until all 1000 students have walked along the row of lockers.

Questions:

1. How many lockers are closed after the third student has passed?
1. How many lockers are closed after the fourth student has passed?
1. After 1000th student have passed, what is the state of the 100th locker?
1. After 100th students have passed, what is the state of the 1000th locker?
1. After 1000th students have passed, how many lockers will still be open?

Question 1:

1st student  =  O O O O O O O O O O O O O O O O O O O O

2nd student =  O C O C O C O C O C O C O C O C O C O C O C

3rd student =  O C C C O O O C C C O O O C C C O O O C C C

Closed = $$3 \times 166 + 3$$ = 501 closed

Question 2:

4th student =  O C C O O O O O C C O C O C C O O O O O C C O C

Closed = $$5 \times 83 + 2$$ = 417 closed

Question 3:

100th locker = $$2^{2} \times 5^{2}$$ = 9 students = odd number = OPEN

Question 4:

1000th locker = $$2^{3}\times 5^{3}$$ = 16 students

but not 125th , 200th , 250th , 500th , 1000th  students

= $$16$$ – $$5$$ = 11 students = odd number = OPEN

Question 5:

Closed = even number of factors (eg 6 = 1, 2, 3, 6)

Open = odd number of factors (eg 16 = 1, 2, 4, 8, 16 or any square)

Open = n2 lockers, for integer n = 1 to 31

By Mike Baxter