Imagine there are 100 lockers. Each locker is given a number 1-100. When the students are assigned to their lockers, the 1st student runds down the row of lockers and opens every door, student #2 closes the doors of locker #'s, 2,4,6,8, etc. to the end of the line. Student #3 changes the state of the doors of lockers 3,6,9,12, etc., student #4 open doors are closed, and closed doors are open. What patterns are developed and which lockers were open at the end, and why were they open?

Question

Imagine there are 100 lockers.  Each locker is given a number 1-100.  When the students are assigned to their lockers, the 1st student runds down the row of lockers and opens every door, student #2 closes the doors of locker #'s, 2,4,6,8, etc. to the end of the line. Student #3 changes the state of the doors of lockers 3,6,9,12, etc., student #4 open doors are closed, and closed doors are open.  What patterns are developed and which lockers were open at the end, and why were they open?

Reiny · Answer

This a classic and very old problem

As a matter of fact, if you google
"locker closing problem" you get quite a few hits.

Here is your problem extended to 1000 lockers.
enter 1 into the window, then 2, then 3 etc. to see that pattern developing.

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On this page there is a good explanation under the heading of "Using locker boards" a bit down the page

http://mathforum.org/alejandre/frisbie/student.locker.html

Here is a clip where some grade 3's illustrate the problem using cards.

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Enjoy