The Locker Problem

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The locker problem on the surface seems relatively straight forward.  My approach to something like this is to visualize it to make it concrete in my mind.  So I took your suggestion to write it out.  In addition, I reduced the size of the problem so it was easier for me to visualize and see any patterns.

With that in mind, I'll use '_' for closed and '|' as open.


Locker # 1 2 3 4 5 6 7 8 9 10
S1 closed - - - - - - - - - -
S2 open - | - | - | - | - |
S3 closed - | - | - - - | - |
S4 open - | - | - - - | - |
S5 closed - | - | - - - | - -
S6 open - | - | - - - | - -
S7 closed - | - | - - - | - -
S8 open - | - | - - - | - -
S9 closed - | - | - - - | - -
S10 open - | - | - - - | - |

However, I can't tell much from this either.  Perhaps I haven't gone far enough in my diagram.  Perhaps if I had stopped at 9 lockers, I would have noticed that for 9 lockers, 3 were left open.  9 is a perfect square with an odd number of factors.   So then by extension, all numbers that are perfect squares are left open at the end of the exercise.

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