Posted by Amy on Thursday, April 26, 2012 at 12:39am.
You need to think in base 3 for this one. Note that using two weights of 1 and 3, you can weigh everything from 1 to 4 grams.
If we list the weights on each side as a pair of numbers, we get
1 = (1,0)
2 = (3,1)
3 = (3,0)
4 = (1+3,0)
If we add a weight of 9, now we can get up to 13
5 = (9,1+3)
6 = (9,3)
...
You can think of it as adding or subtracting base-3 digits. Putting weights on the same side adds them, putting them on opposite sides of the balance subtracts one from the other.
1 = 1
2 = 3-1
3 = 3
4 = 3+1
5 = 9 - (3+1)
6 = 9 - 3
7 = 9 - (3-1)
8 = 9 - 1
9 = 9
So, with weights of 1,3,9,27,81 you can weigh all values from 1 to 121.