There are 1000 lockers in a high school with 1000 students. The problem begins with the first student opening all 1000 lockers; next the second student closes lockers 2,4,6,8,10 and so on to locker 1000; the third student changes the state (opens lockers closed, closes lockers open) on lockers 3,6,9,12,15 and so on; the fourth student changes the state of lockers 4,8,12,16 and so on. This goes on until every student has had a turn.
How many lockers will be open at the end?
I worked out this problem by doing it for 10 lockers. The pattern was obvious (1,4,9 remained open).It's easy to then say that 31 lockers remain open at the end. I don't quite understand why only square numbers remain open at the end. Is there some deeper reason?