The most instinctive way for me to tackle this question was to make a sample size large enough to give me a general picture. I chose 20 lockers. This is the result that I came up with:
At this point I got stuck because I realized I mixed up two things. I did not know whether I meant that the number of lockers that remain “unchanged” was the greatest power of 2 less than the total number of lockers, or if I said locker numbers that were powers of 2 were the ones that remain unchanged (4, 8, 16, etc.). When I looked back at the table I created with the lockers, I realized the lockers that were unchanged were 1, 4, 9 and 16 which are square numbers.
The lockers that remain “unchanged” (end in the same state at which they began) are the square numbers – ie. 1, 4, 9, 16. Let’s assume this to be true, how can we prove that it is in fact the answer?
For a locker to remain “unchanged”, it has to be touched an even number of times (excluding the initial touch by the first student). If we explain this in terms of numbers, it means that the number should have an odd number of factors. It should have an odd number because after we exclude 1 from its list of factors (since we are excluding the initial touch by the first student), it will have an even number of factors left. I mean the answer is apparent, the only numbers with an odd number of factors are square numbers. If we have a square number x, it’s factors are: 1, √x and x.
Hi Sahl, I really like the way you presented all your thoughts, even the parts where you got confused. It shows a thorough problem-solving process. Although the following photos were a bit confusing, your explanations were clear and helped me follow along. Great job!
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