The Scales Problem

We have four different weights to choose from. These will be labelled as a, b, c, d, with each letter be written in ascending order of the weight. Since the vendor sells herbs ranging in weight from 1 g to 40 g, the smallest weight must be 1 g (i.e. a = 1 g). In addition, the sum of the weights must add up to 40 g, such that

a + b + c + d = 40 g

Now, in order to measure 2 g, we could say that the second lightest weight is 2 g. However, another way of doing this is by letting a mass of 2 g be achieved via a weight on both pans. Therefore, a subtraction is needed, and so we have b - a = 2 g. Since a = 1 g, it follows that b = 3 g. With a and b solved for now, I can measure weights of 1 g, 2 g, 3 g, and 4 g. But how can I measure 5 g? Well, the third weight, c, will allow me to. If 5 g is the smallest weight I have yet to obtain, then I must subtract both a and b from c in order to achieve it. So c - a - b = 5, or c = 5 + a + b = 5 + 1 + 3 = 9. This just leaves d now, which can be solved for from the above equation: d = 40 - 1 - 3 - 9 = 27. Thus, the value of the four weights are 1 g, 3 g, 9 g, and 27 g. Looking at these values, these numbers didn't come from nowhere, as it is quite apparent that they follow a distinct pattern: 3^(n-1), where n = weight number. A quick check shows that these four weights account for all the different weights from 1 g to 40 g.

I don't believe there are several correct solutions, as no other way will allow you to account for each weight distinctly, but I could be wrong as I did not delve too deep into proving it.

This puzzle could be extended by looking at different maximum weights (1 g, 4 g, 13 g, 121 g, etc.) and noticing the pattern for which weights you need, and then trying to deduce the formula that tells you the specific weights you need. This could tie in quite nicely with a unit on sequences and series.