Let $S$ be the sum of integer weights that come with a two pan balance Scale, say $\omega_1 \le \omega_2 \le \omega_3 \le ... \le\omega_n$. Show that all integer-weighted objects in the range $1$ to $S$ can be weighed exactly if and only if $\omega_1=1$ and $$\omega_{j+1} \le 2 \left( \sum_{l=1}^{j} \omega_l\right) +1$$