Locker doors and multiples
This semester I am teaching a course for future elementary school teachers. This week we are starting a unit on number theory, and are talking about things like factors and multiples. Here is an extra credit homework problem I plan to give to my students on Wednesday:
A school has 1000 lockers, numbered consecutively from 1 to 1000. The school also has exactly 1000 students. The first student walks into the school and opens every locker door. Then the second student closes the door on every locker labeled by an even number. Then the third student switches the doors on every locker labeled by a multiple of three (that is, opens the door if it is closed, and closes the door if it is open). Then the fourth student switches the doors on every locker labeled by a multiple of four. Then the fifth student switches the doors on every locker labeled by a multiple of five. And so on, for all the rest of the students. After all 1000 students have walked through the school, which locker doors will remain open?
