Example 5 It is known that among 4 coins, there may be counterfeit coins, where genuine coins each weigh 10 grams, and counterfeit coins each weigh 9 grams. Now there is a balance scale, which can weigh the total weight of the objects on the tray. Question: What is the minimum number of weighings needed to ensure that the authenticity of each coin can be identified?
At least 3 weighings can achieve this.
In fact, let the 4 coins be $a, b, c, d$. Weigh $a+b+c, a+b+d$, and $a+c+d$ three times. The sum of these three weights is $3a + 2(b+c+d)$, so if the sum of these three weights is odd, then $a$ is the counterfeit coin; otherwise, $a$ is genuine. Once $a$ is determined, solving the system of three linear equations for $b, c, d$ can determine the authenticity of $b, c, d$. Therefore, 3 weighings are sufficient.

Next, we prove that two weighings cannot guarantee the determination of the authenticity of each coin.
Notice that if two coins, for example, $a, b$, either both appear or both do not appear in each weighing, then when $a, b$ are one genuine and one counterfeit, swapping the authenticity of $a, b$ does not affect the weighing results, so their authenticity cannot be determined.

If there is a weighing in which at most two coins, for example, $a, b$, appear, then in the other weighing, $c, d$ can only have exactly one on the scale (otherwise, swapping the odd/even nature of $c, d$ does not affect the result), at this point, one coin does not appear in both weighings, and changing its authenticity does not affect the weighing results, thus its authenticity cannot be determined. Therefore, at least 3 coins must be on the scale in each weighing, which means there must be two coins that appear in both weighings, leading to a contradiction.

In summary, at least 3 weighings are required.