We are standing in front of 100 lockers arranged side by side, all of which are closed. One man has a bunch of keys with all 100 keys and will pass the lockers exactly a hundred times, opening or closing some of them.
On the first pass, he opens all the lockers. On the second pass, the man will go to every other locker and change its state. That means: If it is closed, it will be opened. If it is already open, it will be closed. In this case, he closes lockers 2, 4, 6… 98 and 100, because all doors were open before.
On the third pass, he changes the state of every third locker – that is, 3, 6, 9, … 96, 99. Closed doors are opened, open doors closed. In the fourth pass, every fourth locker is changed, at the fifth every fifth – and so on. At the last, the 100th, the man finally only changes the state of door number 100.
The question is: How many of the 100 compartments are open after the 100th pass?
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