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If we are given a set of $n$ weights and a balance we can weigh $2^n-1$ different sized objects by putting them one side of the balance and comparing them with $2^n-1$ set of weights. However we can also find object weights by putting it and some known weight on the right side and balancing them with known weight of left side. What is the greatest number of different amounts we can measure this way ?

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$$\frac{3^n-1}{2}$$

Consider $n$ weights of value $1,3,9,...,3^{n-1}$. By putting those weights on both sides, you will be able to compare any object weight to any value between $1$ and $\frac{3^n-1}{2}$

Example to compare object $A$ to $5$ with $3$ weigths :

$$\underline{\phantom{3,}9\phantom{,3}}\quad\uparrow\quad \underline{3,1,A}$$

This is an optimum due to the fact you have 3 possibilities for each weight (left, right, none), so you write in base 3 with (1,-1,0), and can obtain any number between $-\frac{3^n-1}{2}$ and $\frac{3^n-1}{2}$, that is exactly $3^n$ different numbers.

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