Response to "Market Scales Puzzle"

When attempting the main question, I tried a whole bunch of other numbers based on some innate interpretation of "understanding" (guess and test), but none of them worked out. It was also both relieving and concerning that the solution is probably unique. On one hand, it's nice that there is probably only one answer, so we can all agree on it -- it's a hint in itself. On the other hand, it feels so restrictive, knowing that only four out of the many integers could possibly work! I also remember getting tangled up thinking about how subtraction is also fair game (placing the weights on the herb side), which made me stumble around. "Would 1 be 1? 3 - 2? 5 - 4?"

In my frustration, I took a detour to the other problem (5 weights, up to 31 grams) since the problem of subtraction wasn't there anymore. I wised up quickly to the fact that 31 is a suspiciously specific number, in addition to being given 5 weights. "Let's give one weight the value of 1 since it's the only way we can get 1. Now we need to get 2. Oh, we need a weight of 2. No other way to do it." I also connected the dots when I saw how many combinations of weights could be possible, so very quickly I realized it was just a binary kind of deal up to 5 digits, so I concluded that the 5 weights should be 1, 2, 4, 8, 16.

I came back to the original question, equipped with the base-wise thinking that helped me solve the problem above... only to fall back to guess and pray. That did not work, so I got desperate and reread the question. As you can see from the picture below, you can see that I used the hints given to guide myself back to base-wise thinking:

I quickly realized the going with base 2 wouldn't be productive, since 4 digits of it would only go up to 15 (1, 2, 4, 8). So I (once again) guessed and tested, this time with base 3. As I went along the target numbers, everything worked out between the base-3 numbers and the constraints of the problem! Thus, my answer to the original problem is 1, 3, 9, 27.

Though I did end up solving the problem with indisputable proof, I will admit that there was a LOT of guessing, testing, and praying. This is something that I hope to address, especially since there is a beauty in terms of bases happening here, along with how they intertwine with the basic addition and subtractions of the problem. As such, my first order of business would be to guide students along the same path that I traversed: undergo some initial immersion in the world of ancient Egyptian mathematics, introduce them to number base systems, and give them the one-pan-scale 5-weights problem first (or even a simpler version where they only need to go up to 15 grams, and only 4 weights are allowed). From this problem, a possible extension after the initial questions could be to generalize. "If we had n weights, what's the highest number of grams we can go while capturing all possible values between 1 and the highest number?" I could ask, with the obligatory request for students to explain, "Why is that?"

Another challenging extension that could be interesting is to lift the restriction that the herbs cannot be used as auxiliary weights. What if you were allowed to use a previously weighed bundle of herbs once? For example, if I had the weights 6g and 4g, I could produce a 2 gram bundle of herbs. From there, I could use the two-pan scale to weigh out an 8 gram bundle (by using the 6g weight and the initial 2g of herbs), and then I cannot repeat this recursive operation again for this instance of weighing. Although it wouldn't be too useful to help students understand (since I'm not even sure if it abides by a pattern of number bases), it could prove to be a fun question to pose to them, to get them thinking mathematically.

To address the last point, this puzzle is definitely driven by my understanding of how base systems work, considering that the solutions to each of the 2 questions were a geometric sequence starting from 1. There's also some interesting patterns that could be gleamed from the solution, if we apply a number theory way of thinking. From 5 to 13, we can notice:

• 9 is in the middle of 5 and 13, and is 4 apart from each.
• The range of 5 and 13 can be created by taking 9, and adding/subtracting some combination of 3 and 1. Possible combinations:
• 0, 0 --> 9
• 0, +1 --> 10
• 0, -1 --> 8
• +3, 0 --> 12
• +3, +1 --> 13
• +3, -1 --> 11
• -3, 0 --> 6
• -3, +1 --> 7
• -3, -1 --> 5
• Observation: there are 9 possible combinations. Each place (+/-/0 3, +/-/0 1) can take a total of 3 states, which matches up with 3^2 (which would be the 2nd place value). The sum of the smaller numbers only goes up to 4, which is also in line.

• Observation: a similar effect can be observed from 27 and the range 14 to 40.
• The largest sum of the smaller numbers is 9 + 3 + 1 = 13, which lines up with the gap between 27 and the range bounds.
• For each of those smaller numbers, there are 3^3 possible combinations of (+/-/0 9, +/-/0 3, +/-/0 1), or 27 possible combinations. The combinations together fill in all numbers 14 to 40.

• Extension idea: If we had 5 weights, what's the highest number we can achieve? Going off the pattern observed above, it would be 3^4 + (3^3 + 3^2 + 3^1 + 3^0), or 81 + 40 = 121.
• If we had n weights, how high can we go? Why is it possible?
• Such extensions aim to build pattern recognition within the context of number theory and base systems.

Of course, there are also questions that can be raised further, like what accommodations we'd need in order to make a base 4 weight system work. Would we need to allow re-weighing as described above? Would we need to allow using each weight number at most twice? These ideas are effectively hypotheses to address the fact that the gaps would be too large (ex. 1 to 4 cannot simply be done with weights 1 and 4; how do we create 2, for instance), and they could serve to guide our extensions as well in order to enrichen student understanding and inquiry into number theory.