The Market Scale Problem

This problem was quite challenging for me, maybe partially because I’ve only taken one number theory class and I didn’t really find I connected with it. Either way, I was able to find a solution to the problem, but almost completely by trial and error. I’ve attached my work because I still think the way I worked through the problem is interesting, but I unfortunately don’t feel I have a strong grasp as to why the weights must be 1, 3, 9, and 27. I see that they are all powers of 3 (and this gives me the intuition that the idea relates to bases), and I know that each combination of adding and subtracting these 4 numbers had to be unique in order to get every number from 1 to 40, but unfortunately I haven’t been able to make sense of the solution beyond this. Additionally, when I write out the math, I can see the pattern emerging of how the numbers interact and it seems intuitive in some way to me that we can use these numbers as a base to create numbers 1 through 40. However, I haven’t been able to find a way to describe this pattern mathematically.

Similarly, after brute-force solving this question, the follow up question about 5 weights and a single-sided scale was much easier to guess and to understand. Just recognizing that 2^0+2^1+2^2+2^3+2^4 = 31 and that the last question had to do with powers of 3, I was able to figure out quickly that the weights would have to be 1, 2, 4, 8, 16. Further, I know that I can write any number in base 10 as a combination of powers of 2, so it was easy to reason that my guess worked without having to test every number like I did in the original problem posed. I think the difficulty going between this one and the original question is that I’m not sure how to fit subtracting numbers rather than strictly adding into it all.

I think puzzles like this are fun to bring into a math classroom just to explore fun math and gain some number sense and appreciation for the applications of math without the associated pressures of a regular lesson or unit. Once again, since I wasn’t able to reason out the original problem, it feels hard to find a way to extend it to a classroom setting. The one idea I have relating to the follow up problem is using this in a history of math class as a way to motivate the idea of base 2 (and other bases) and developing some sense of transferring between bases before giving it a name and introducing it as conversion between bases. I think this gives students a tool to constantly reference as they are learning about bases and dealing with something that probably feels abstract and new to them.





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