Using the Locker Boards
Step OneStudent One opens all of the lockers.
Step TwoStudent 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 ThreeStudent 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 FourStudent 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 FiveStudent 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 SixStudent 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 SevenStudent 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 EightStudent 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 NineStudent 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 TenStudent 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:

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