Consider a balance scale with 2 pans, $A$ and $B$. Show that any weight not exceeding $\left(3^{k}-1\right) / 2$ that is placed in pan $A$ may be measured, by placing in either pan $A$ or $B$, a subset of weights of $\left\{1,3,3^{2}, \ldots, 3^{k-1}\right\}$.