Market Scale Puzzle

Going through elimination (and with a hint from Saiya), the four weights have to be 1, 3, 9 and 27 grams. We can weigh up to 4 grams with the weights 1 and 3. We can weigh up to 13 grams using the weights 1, 3 and 9. We can weigh up to 40 grams using all 4. The weights would have to be used in a variety of ways such as weights on both sides, omitting some weights (ie. for 30 grams we would only use the 27 and 3 gram weights), etc. I was talking to TsáKtalay’pa a little while after receiving this problem and he said he didn't use the weight 3 (if I remember correctly) so the wording in the question "must" is interesting - I wonder if there is only one combination or if there are multiple ways to weight 40 grams. 

On a one pan scale you would need the weights 1, 2, 4, 8 and 16 to weigh up to 31 grams. This problem was surprisingly much more straight forward since there was only one pan that can be used to weigh. As soon as you maxed out the amount of weight, you knew the next weight needed - for example with the weights 1, 2 and 4 you can weigh up to 7 grams and so you know the next weight needed has to be 8. 

I feel like the two pan scale problem is really counter-intuitive in the beginning. Although I can't think of any explicit connections to the secondary curriculum I do believe it could induce a larger theme around exploratory mathematics. If students took this on in a classroom activity with a real two pan scale and different weights, I strongly believe it could challenge their initial perceptions about how to combine numbers. It's almost like a physical manipulation of left side equals right side, and all the different ways that weights (terms) can be moved around and utilized to have both sides be equal.