The market scales puzzle

 The values of the 4 weights must be 1, 3, 9, 27! The first 4 powers of 3.

I've never used a two-pan scale, and I'm mostly aware of how it works, but I couldn't quite wrap my head around the weights I would need. So I started with the question about a one-pan scale. That one felt very intuitive. I would need the weights 1, 2, 4, 8, 16. Those add up to a weight of 31, and you can create any weight from 1-31 with just those numbers. From there, I mulled over the two-pan scale, trying to make it work with different powers of 2. However, no matter how I played with the numbers, it didn't seem to quite work to make 40 with just 4 weights. I tried powers of 4, and those numbers went way too big! So, continuing with the idea of powers, I tried powers of 3, and it added up to 40, which was a good sign. And testing out different potential weights, it worked. I realized this worked because there's only 3 possible positions each weight can be in, whereas with the one-pan scale, theres only 2. 

I think this problem could be extended to different weight numbers and numbers weights. Students could also compare one-pan vs two-pan scales and investigate why powers of 2 work for one, while powers of 3 work for the other. This connects to number theory and bases because the weights are powers of 2 or 3, depending on the scale type. The one-pan scale relates to base 2, where each weight is either used or not used, while the two-pan scale relates to base 3, where each weight can be placed on the left, right, or not used!