Larry Barrett
Using a spreadsheet, I pushed the simpler test cases from 1 student and 1 locker to 25 students and 25 lockers. Here is what I observe (assuming no mistakes):
For the cases 1-1, 2-2, 3-3 the end result is 1 locker open (3 cases)
For the cases 4-4 to 8-8, the end result is 2 lockers open (5 cases)
For the cases 9-9 to 15-15, the end result is 3 lockers open (7 cases)
For the cases 16-16 to 24-24, the end result is 4 lockers open (9 cases).

It looks like it is possible to extend this and I would expect to see, for instance, for the next 11 cases (25-25 to 35-35) the end result would be 5 lockers open, etc.

If this works out, I would try to find a formula to extend this to the 1000th case.
Weather is finally warming up here in MD so time to get outside and do some work.