When this problem was presented to me, I became very anxious and was reminded of the first proofs class I took at UBC that changed my entire idea of Math as a discipline. But like all problems, the best way to start is to break it into manageable chunks. Firstly, I drew a sample diagram of 10 lockers and simulated the problem on a small scale. At that point, I found that lockers 1, 4, and 9 were open. I suspected the open lockers would be perfect squares, so I simulated a 20-locker problem and found that locker 16 was also opened. Now, I felt fairly confident I was on the right track with the perfect square idea.
But why? This next connection took my slightly math-rusty brain a while to make. What does it mean that a number is a perfect square? Well since all factors of composite numbers come in pairs and perfect squares have pairs with the same number, this means perfect squares have an odd number of factors. In the locker problem, the number of factors is how many times a locker will be touched. If the original state of all lockers is closed, an odd number of touches would mean the locker is open at the end of the simulation. Therefore, lockers with a number that is a perfect square will be open at the end of the simulation!
Could you please explain more about how you figured out the connection to perfect squares when you worked on lockers 1 to 10? I noticed that there were still other lockers open, so I'd love to understand how you concluded that perfect squares were the key.
ReplyDeleteHi Malihe, I'm not quite sure what you mean. When I ran the simulation for 10 lockers, only lockers 1, 4, and 9 were left open (denoted with open circle; closed lockers have an X). I tried to think about what these numbers had in common and suspected that perfect squares was the connection. Then, I decided to run the simulation again with 10 more lockers to test my theory, and of the next 10 lockers, only #16 was open. By that point, I knew the answer was perfect squares! Does that clarify some on my work? Thanks!
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