The chemistry teacher at Stevenson High School is ordering equipment for the laboratory. She wants to order sets of five weights totaling 121 grams for each lab station. Students will need to be able to weigh every integral weight from 1 to 121 grams using these weights and a two pound balance. What weights should the teacher order?
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.