, 2010

The possibility to determine mass $m$ means the possibility to place the weights on the two scalepans so that the difference of total masses on the two scalepans is exactly $m$.

Every weight can be placed on either of the two pans or on neither of the pans. For $n$ weights this makes $3^n$ different placements. Note that the placement where none of the weights is on the scales does not determine any mass. Also, for each placement there is a symmetric placement with all the weights on the two pans swapped, which determines the same mass. Therefore with $n$ weights it is possible to determine at most $\frac{3^n-1}{2}$ different masses.

We show by induction that it is possible to determine all masses from $1$ to $\frac{3^n-1}{2}$ using $n$ weights with masses $1$, $3$, $\dots$, $3^{n-1}$.

For $n=1$ it is obvious. Assume that the claim holds for $n=k$. By the induction assumption we can determine all masses from $1$ to $\frac{3^k-1}{2}$ by weights $1$, $3$, $\dots$, $3^{k-1}$. Using the weight with mass $3^k$, we can determine the mass $3^k$, and using it together with the other weights also the masses $3^k+1$, $\dots$, $3^k+\frac{3^k-1}{2}$ and $3^k-1$, $\dots$, $3^k-\frac{3^k-1}{2}$.

Since $3^k - \frac{3^k-1}{2} = \frac{3^k-1}{2} + 1$ and $3^k + \frac{3^k-1}{2} = \frac{3^{k+1}-1}{2}$, the claim is also true for $n=k+1$.