There are 100 doors in a row that are all initially closed.You make 100 passes by the doors.The first time through, visit every door and toggle the door (if the door is closed, open it;if it is open, close it).The second time, only visit every 2nd door(door #2, #4, #6, …),and toggle it.The third time, visit every 3rd door(door #3, #6, #9, …), etc,until you only visit the 100th door.
Answer the question: what state are the doors in after the last pass? Which are open, which are closed?

# Solution

n + n/2 + n/3 + n/4 + … + n/n
= n(1+1/2+1/3+1/4+…+1/n)
= n(ln(n+1)+r)