This one took a while, but the weights would be 1, 3, 9, and 27 (I have attached my work below). I first started with 1, 2, 4, 9, and then realized that I can balance on both sides, and so after a bit of trial and error, and a LOT of patience, I was able to find the answers. The pattern here is that they are all powers of 3, which sort of makes sense. Give that we can use the weights of the herbs, and that we can place weights on the herb’s side, we can hit all the numbers in base 3 because we can add and subtract from the herb’s weight.
This relates to the second problem, and seeing the weight 31 immediately made me connect that the weights would be in powers of 2, and including 1. So, the weights would be 1, 2, 4, 8, and 16. I think that it’s neat I can find the solution very quickly after doing the previous question, which shows me that I have relational understanding, at least a bit. To double check, 1+2+4+8+16 = 31, which makes sense!
With this, I can ask the students how many grams of herbs can 5 weights measure in a two-panned scale. If there were 5 weights, we can go up to 1+3+9+27+81 = 121 grams of herbs! Or I can ask the students what combinations of weights would I need to measure out 27 in a one-panned scale with the “powers of 2 weights.” It would be 16+8+2+1 = 27, and right there, I just taught them binary system without showing them the formula for calculating bases!
As I am writing this, it’s already blowing my mind. Of course, that’s who we visualize bases! I have never even considered visualizing the concept of bases, yet this is so simple and kind of fun! I found the solution while I was driving, so I had to pull over, write the idea down, and keep driving so I don’t forget. Honestly, it’s a very fun feeling, and I would love to imagine my students just suddenly figure out a hard question while they are going about their day. I don't understand number theory very well, so I can’t explain why the base 3 problem works, but I have some sort of understanding that if I have a 81 g weight, I can measure all herbs of up to 121. It’s a weird feeling, but honestly something I would want my students to experience too. Definitely saving this exercise for them :)
What a delightful post to read, Leon! I'm thrilled that you had to pull over to write down your ideas as they came to you while driving -- isn't that what the excitement of mathematical thinking and discovery is all about? Great solution and very thoughtful comments!
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