13.25 Given 4 coins, one of which may be counterfeit, each genuine coin weighs 10 grams, and the counterfeit coin weighs 9 grams. Now there is a balance scale with one pan that can measure the total weight of the objects on the pan. To identify whether each coin is genuine or counterfeit, what is the minimum number of weighings required?
[Solution] Let the weights of the 4 coins be denoted as $a, b, c, d$. Weigh them 3 times: the first time, weigh $S_{1}=a+b+c$, the second time, weigh $S_{2}=a+b+d$, and the third time, weigh $S_{3}=a+c+d$. Thus, we have
$$
S=S_{1}+S_{2}+S_{3}=3 a+2 b+2 c+2 d .
$$

From the parity of $S$, we can determine whether $a$ is 9 or 10. Since
$$
S_{2}+S_{3}-S_{1}-a=2 d,
$$

we can find the value of $d$. Consequently, we can obtain the values of $b$ and $c$ from $S_{2}-a-d$ and $S_{3}-a-d$. Therefore, weighing 3 times can identify the authenticity of each coin.
If only two weighings are performed, it is insufficient:
(1) If the first weighing involves only 1 coin, the second weighing cannot determine the authenticity of the other 3 coins;
(2) If the first weighing of two coins totals 19 grams, the second weighing cannot simultaneously resolve the authenticity of all 4 coins; $\square$
(3) If the first weighing of 3 coins totals 29 grams, the second weighing, whether it involves a part of the 3 coins or the 4th coin, cannot simultaneously resolve the authenticity of all 4 coins.
In summary, a minimum of 3 weighings is required.