Weighing Accurately

The Problem:

What is the smallest set of weights needed in order to weigh every integral number of pounds up to 255 pounds by putting the weights on one side of a balance scale and the object to be weighed on the other side?

If you are allowed to put weights on both sides of the scale, what is the smallest set of weights needed for the above problem?

 

Solution:

To represent the one side case you use the powers of two as weights. This would require the weights 1,2,4,8,16,32,64 and 128. Which weights you use would look a lot like the binary representation for the number.
The second case you get three bits of info per number so you can use powers of threes as weights. This means the weights 1,3,9,27,81 and 243g. For example to weigh something that is 22g you would put the 27, 1 and 3g on the unweighted side and the 9g on the side with the unknown object to identify it. Fun!

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