Math puzzle- Scale and reflection

I have thought about this question for a while,  I think the numbers should be spread between 1 and 40 and they should be added up to 40, so I can cover all the numbers from 1 to 40. and I decide the weight 1 and 3 are needed, so I can weigh 1, 2( with 1 on the other side), 3, and 4. Now I need to decide my 3rd number which is can not be too close to 3 but cannot over 17 since it will be over 40 if we add the fourth number. I couldn’t see the pattern right away, but I was sure that there must be a pattern for this problem.  I did lots of trial-and-error methods and I decide to try the third number as 9 and the fourth number is 27 since the first two numbers have a ratio of 3. luckily, they add up to  40. Then I start to test if the four weights are 1, 3, 9, and 27 works for this puzzle.

use 1- measure 1

use 1,3- measure 2 ( with 1 on the herb side)

use 3-measure 3 

use 3 and 1- measure 4

use1,3,9 -measure 5(1,3 on herb side)

use 3,9- measure 6 ( 3 on herb side)

use 1,9-measure 10

use 1,3,9-measure 11 ( 1 on the herb side)

use 3,9- measure 12, I will use arithmetic to illustrate for the rest.

1+3+9=13, 27-(1+3+9)=14, 27-(3+9)=15, 27+1-(9+3)=16, 27-(9+1)=17

27-9=18, 27+1-9=19, 27+3-(9+1)=20, 27+3-9=21, 27+1+3-9=22, 27-(1+3)=23

27-3=24, 27+1-3=25, 27-1=26, 27, 27+1=28, 27+3-1=29, 27+3=30, 27+3+1=31

27+9-(1+3)=32, 27+9-3=33, 27+9+1-3=34, 27+9-1=35, 27+9=36, 27+9+1=37,

27+9+3-1=38, 27+9+3=39, 27+9+3+1=40, here we go!

I don't think there has more combinations at least I don't find another one.

I can extend this puzzle to ask students to discuss why these combinations work.

what is the mathematical reason.