Scales Problem

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My first thought when I read this problem was that one of the weights must be equal to 1 gram. This must be true because in order to weigh 1 gram of herb, there must be a weight that is 1 gram. My next thought was that to weigh 40 grams of herbs, I should probably use all the weights to add up to 40 to be most efficient.  I thought that I would probably need a 2 gram weight to weigh the even numbers. But then I thought, could I be more efficient and instead 'make' a 2 gram weight by combining a 3 gram and 1 gram weight? This would be more efficient cause I could also combine 3+1 to make 4.

So far I have determined that there must be a 1 and 3 gram weight. This will make it possible to weigh herbs of grams 1 to 4. To weigh a 2 gram herb, we put it together with the 1 gram on one side, then check it with the 3 gram on the other side. So now that I can make 1 to 4, how can I make 5? By this same reasoning, I chose a 9 gram weight. This would allow me to do 9-3-1=5. This gave me a bigger max range of 9+3+1=13. I also checked that I was now able to make any combination between 1 to 13. For example with 11 gram herb, I would put it with 1 gram on the left side, then the 9 and 3 gram on the right side, this makes 12=12. By this same logic I am finding out how to make the next number I need which is 14. We can use algebra X-9-3-1=14, which gives us X=27. The 4 weights should be 1,3,9, and 27 grams. These combinations will be able to weigh anything from 1 to 40 grams.

I haven't explored any other solutions but I found this method to be intuitive and easy to understand. I could probably extend this problem by showing them this application in real life. This will help them understand the beauty of the solution even more. You could also probably change the numbers in order to make it easier or harder.

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