Hello Everyone,
First I started with drawing out the problem to see if I noticed any patterns.
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| Figure 1 |
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| Figure 2 |
After I realized that lockers do not have a change in state after they have been passed, I considered prime numbers. Prime numbered lockers will only be touched by Student 1 and itself. Hence, each locker is first closed by Student #1, and is then reopened by their respective numbered student.
Therefore, I was able to conclude that thus far, locker 1 will be closed, and all prime numbered lockers will be open.
The next objective I wanted to consider was how odd and even numbers were affected.
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| Figure 3 |
I considered 6 because I already saw that 4 finished closed, so I believed perhaps even numbers finished closed. Unfortunately, this did not yield any helpful results. Moreover, I was able to further infer that odd numbers would not particularly yield much of a different outcome as well.
However, it was this conjecture that brought my attention to a number’s factors.
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| Figure 4 |
I noticed that numbers that had an even number of factors finished open. This led me to my next theory: lockers with an even numbers of factors will finish open, and lockers with an odd number of factors will finish closed.
To test this, I arbitrarily chose numbers 8, 12, and 16.
By listing their factors, I indeed did confirm that lockers with an even numbers of factors finished open, and lockers with an odd number of factors finished closed.
Furthermore, as each locker will only be touched by its factors, it logically follows that if touched an even number of times, it will retain its starting state, (open,) and the opposite if touched an odd number of times, (closed.)
Therefore, I conclude that my answer for the Locker Problem is: all lockers with an even numbers of factors will be open, and all lockers with an odd number of factors will be closed. As the question does not ask for a specific amount of lockers which are opened or closed, we are done.
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