Scales Problem

 After fumbling through a few ideas of how to start solving this puzzle, I decided on the method of  picking four numbers that add to 40, that can account for all amounts between 1 and 40. However 40 is quite a large number so there would have to be at least one larger number, likely over 20 along with three others that will help add to the lower numbers between 1 and 40.

So first I tried to see if a combination of 4 numbers including 20 could work. With 20 the three other numbers to add to a total of 40 were 2, 3 and 15. However I found numbers with a multiple of 5 were hard to work with when attempting to account for all amounts between 1 and 40. Thus my next step was to try numbers that did not have 5 as a factor. As well, I chose to keep 3 in the combinations of the numbers as it seemed very useful and could be used to build on other weights to get the amounts from 1- 40. I also thought I would try and use 1 in the four weight combination as having 3 and 1 together are also useful in getting amounts between 1 and 40. So with 1 and 3 the other numbers that I picked were 27 and 9, which are again multiples of 3. I decided to try and use these numbers to weigh out each amounts of herbs. Using the table below I found that the four weights, 1, 3, 9 and 27 were the weights to be able to account for all amounts of herbs between 1 and 40.

I am fairly certain there are no other correct solutions, however as I solved this puzzle with a guess and check method I really have no proof that there is no other answer. I did however try a lot of number combinations that I did not elaborate on, so with the number of combinations I tried I do not think there are any other correct solutions. 

I believe a way to extend this puzzle to help students understand the mathematics more deeply would be to reduce the range of amount of herbs or some other substance to around 20 so there would not be as many combinations of numbers that students would have to sift through that add to 20 but also can be used to equal out each side of the scale with the product on the scale. I believe doing this would allow students to do less of a guess and check method and instead reason with the combinations of numbers that add to 20 and see what number properties such as factors would be best to accommodate all amounts from 1-20.