Weight scale reflection
The first connection I made to this problem was that I most likely would not discover a formula that would work, yet I figured there would be a pattern that should be able to be written as a formula. That being said, the first problem was of higher complexity given that you could use both scales to get from 1-40 so I tried guess and check.
I actually started working my way back and forth by increasing from 1 and decreasing from 40. Obviously, we needed a 1. Then for 2 either we would have to pick a 2 or we could use a 3. Given that we can use both scales I think it would be more beneficial to use the 3 since it would also satisfy the next term and we can generate 2 from 3-1. At this point I looked at 40 and thought about two options, first 30 & 6 and then 27 & 9. The reason 30 & 6 failed is because I realized that the gap was significant so while it works for numbers close to 40, it fails late teen numbers (like 19) since 30 - (6 + 3 + 1) only gives 20. So then I continued on in the other direction using 9 and quickly realized that both 9 and 27 were powers of 3. So I continued and realized that it worked for all numbers up till 40. So the numbers are 1, 3, 9, and 27
For the other problem, I immediately thought the gap had to be smaller since we can only use one side of the pan. Therefore, I tried powers of 2 and after trying the first 8 numbers, I skipped over and tried some numbers in succession towards 16 and realized that the 5 numbers were 1, 2, 4, 8, and 16.
I would probably not assign this as a problem without getting students started as it is very long (although not complicated). However, an adaption/extension may be to get students to see how high they could go using both pans with powers of 3 and only 4 numbers and then compare it to the power of 3 problem.
The extent of my number theory knowledge is limited but it reminded me of a problem I saw before where they would ask us to generate numbers that satisfy constraints and we had to involve gcf to solve it. Since there were many constraint it involved quite a bit of guess and check before establishing a the smallest number that would (I believe it was from peter liljedahll). This problem reminded me of his problems quite a bit.
Thanks for this good solution and commentary, Zain! And yes, I think that Peter Liljedahl is always looking for interesting problems of this kind for Thinking Classroom applications.
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