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?

Answer 1

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