At first, I thought maybe the weights would be in base two since computers use base two to describe all numbers. However, after doing some trial and error it did not seem possible to get 4 numbers to sum all numbers 1 to 40.
I then switched to equations by letting each number be a variable a, b , c, d. Assuming that the weights could be added to both sides of the scale and that one of the weights had to be 1 gram, I proceeded to try and solve the equations. After doing a few equations for 40 - 34, I noticed either we needed one of the weights to be 3 grams or the difference between two of the weights had to be 3. Since it was easier to assign one of the variables to 3, I decided to continue my calculations with two variables and two known values. Eventually, I got to a point where the two other variable needed to sum to 36.
I began to narrow down the possibilities by trying to find values that are not possible for each pair that sum to 36. One pair that I struggled to find any discrepancies was (9, 27). I confirm that this was a solution, I found representations for each value 1 to 40 using 1, 3, 9 and 27 which ended up working.
Are there several correct solutions?
I'm not sure if there are several correct solutions but I did notice that this solution had values that were all powers of 3.
How could you extend this puzzle to help your students understand the mathematics more deeply?
We could use an actual balance scale to physically see what is happening and how the scale would balance. We could also try to see if we could do this same exercise except with more weights for a larger range (1-100 grams perhaps?) and see if we could use powers of 3 or if the 40 grams was just a special case. We could see if there are other solutions if we allow the number of weights to increase and then discuss why merchants would want as few weights as possible.
Cleaned up version of my rough work:
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