Question 718732
Number the lockers from 1 to 1000.
The problem has to do with the number of divisors of a locker number.

For example; locker number 12.
The divisors of 12 are: 1, 2, 3, 4, 6, and 12.
The locker was opened by the 1st student,
. . then closed by the 2nd student,
. . then opened by the 3rd student,
. . then closed by the 4th student,
. . then opened by the 6th student,
. . then closed by the 12th student.

We can make a general rule.
If the locker number has an even number of divisors,
. . the locker will be closed.

Do any numbers have an odd number of divisors?
We find (perhaps by accident) that squares have an odd number of divisors.

For example: 16 has five divisors 
. . . . . . . . . .25 has three divisors 

Therefore, the lockers that are left open are:
. . 
1, 4 , 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961.