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November 27, 2015
Posted by **Amy** on Thursday, April 26, 2012 at 12:39am.

- calculus -
**Steve**, Thursday, April 26, 2012 at 10:59amYou 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.