20.14. Among $2 n+1$ coins, there are $2 k$ counterfeit ones ( $k \leqslant n$ ), and the weight of a counterfeit coin differs from that of a genuine coin by 1 gram. How can one determine in a single weighing on a balance scale with a pointer whether a chosen coin is counterfeit or not?
20.14. Let's set aside the chosen coin and place half of the remaining coins on one pan of the scales, and the other half on the other pan. Let there be $k_{1}$ counterfeit coins among the first $n$ coins, and $k_{2}$ counterfeit coins among the other $n$ coins. Then, the weight on the first pan is $n a \pm k_{1}$, where $a$ is the weight of a genuine coin, and the weight on the second pan is $n a \pm k_{2}$. The pointer on the scales will show the difference $k_{1}-k_{2} \equiv k_{1}+k_{2}(\bmod 2)$. Thus, if the chosen coin is counterfeit, the pointer reading will be an odd number, and if the chosen coin is genuine, the pointer reading will be an even number.