Weights a Scale and Integral values 1 to 40

Weights a Scale and Integral values 1 to 40 in whatever units you need

The problem posed is to use only four weights to measure from 1 to 40 on a typical twin pan balance scale.

1. The first premise was to add weights to get all the values but with a quick addition this was not possible. How to get 1, by a weight of 1 or a difference in two weights? I chose to start with weight #1 being a weight of one, which leaves 39 numbers remaining. Factor as 13 x3.
2. The key premise was to realize that you could reach some numbers by adding to the bottom number and others by subtracting from the top number. Simply put, the weights could be put on both pans and the difference would allow you to reduce weight from the top number. Thus if the first three weights add up to 13 then the next weight minus 13 needed to be the next number 14. 27-13=14, and you can add the 13 to the 27 weight to get 40.

3. Checking how to get 2 as 3-1=2. Then the second weight must be 3 and the third weight is 1+3+w3=13 or w3 =9. 3+1=4, 9-4=5, 9-3=6, 9-3+1=7,9-1=8, 9, 9+1=10, 9+3-1=11, 9+3=12, 9+3+1=13. We can get all the numbers up to 13.

4. We need 14 which is 40-14=26 and divide by 2 = 13 which can be added or subtracted to get from 14 to 40.

5. This can continue on with n= weight number then;
Weight n = 3^(n-1)
n with a range of
weight 1 = 3^0 = 1 1
weight 1 = 3^1 = 3 1 - 4
weight 1 = 3^2 = 9 1 - 13
weight 1 = 3^3 = 27 1 - 40
weight 1 = 3^4 = 81 1 - 121
weight 1 = 3^5 = 243 1 - 365

and could continue.