Using the Locker Boards

Step One

Student One opens all of the lockers.

Step Two

Student Two closes locker #2 and continues by closing every other locker (every multiple of 2 - 2, 4, 6, 8, 10, 12, 14, 16, 18, 20...)

Step Three

Student Three starts with locker #3 and because it is open, closes it, thereby changing its "state." Student Three continues to change the state of every third locker (every multiple of 3 - 3, 6, 9, 12, 15, 18...)

Step Four

Student Four starts with locker #4 and because it is closed, opens it, thereby changing its "state." Student Four continues to change the state of every fourth locker (every multiple of 4 - 4, 8, 12, 16, 20...)

Step Five

Student Five starts with locker #5 and because it is open, closes it, therefore changing its "state." Student Five continues to change the state of every fifth locker (every multiple of 5 - 5, 10, 15, 20...)

Step Six

Student Six starts with locker #6 and because it is open, closes it, thereby changing its "state." Student Six continues to change the state of every sixth locker (every multiple of 6 - 6, 12, 18...)

Step Seven

Student Seven starts with locker #7 and because it is open, closes it, thereby changing its "state." Student Seven continues to change the state of every seventh locker (every multiple of 7 - 7, 14...)

Step Eight

Student Eight starts with locker #8 and because it is open, closes it, thereby changing its "state." Student Eight continues to change the state of every eighth locker (every multiple of 8 - 6, 12, 18...)

Step Nine

Student Nine starts with locker #9 and because it is closed, opens it, thereby changing its "state." Student Nine continues to change the state of every ninth locker (every multiple of 9 - 9, 18...)

Step Ten

Student Ten starts with locker #10 and because it is open, closes it, thereby changing its "state." Student Ten continues to change the state of every tenth locker (every multiple of 10 - 20...)

At this point you should be able to see which of the 20 doors that you are manipulating will be open and which will be closed. Doors 1, 4, 9, 16, .... will be open. Doors 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, and 20 will be closed. The lockers will look like this: