I have a variant of the locker problem, which I think is more difficult to solve:

There are 1000 students and lockers in a school. At the start is each locker open or closed, at least one of the lockers is closed. Student 1 starts by closing his locker if it was open, and opening it if it was closed. He continues with locker 2, but only if he opened his own locker and so on, i.e. he continues to open lockers until he has to close one. Student 2 do the same procedure, starting with locker 2, and so on until all 1000 students have opened/closed the lockers. Each student will open x (x=0-1000) and close 1 locker. If a student reaches locker 1000, he continues with locker 1.

The problem is to prove that after student 1000, the lockers will be in the same state (opened or closed) as at the start, no matter what this state was (except that at least one of the lockers was closed).

Göran Wicklund