Among 13 completely identical coins, there is one fake coin whose weight does not match the others. What is the minimum number of weighings required to determine which coin is the fake using a balance scale?
Divide the coins into groups of 4, 4, and 5. Let's denote these groups as $a, b, c$. First, place $a$ and $b$ on the scale. If the scale balances, we can assume that $a$ is the lighter. In this case, the counterfeit coin is either in $a$ and is lighter, or in $b$ and is heavier.

In this scenario, assign a number to each coin in the groups. For the second weighing, compare the group consisting of $a_{1}, a_{2}, b_{1}$ with the group consisting of $a_{3}, a_{4}, c_{1}$. If the scale balances, then the counterfeit coin is among $b_{2}, b_{3}, b_{4}$. Weigh $b_{2}$ and $b_{3}$. The heavier one is the counterfeit, and if they balance, then the third one is the counterfeit.

If the first group is lighter in the second weighing, then the counterfeit coin is among $a_{1}$ and $a_{2}$. If the first group is heavier, then the counterfeit coin is among $a_{3}, a_{4}, b_{1}$. (Since if the counterfeit is in group $a$, it must be lighter, and if in group $b$, it must be heavier.) In both cases, weigh the two $a$ coins. The lighter one is the counterfeit, and if they balance, the remaining suspicious coin from $b$ is the counterfeit.

If the counterfeit coin is in $c$ based on the first weighing, then for the second weighing, place two coins from $c$ on one side of the scale and one coin from $c$ plus one genuine coin on the other side. If the scale balances, then for the third weighing, compare one of the remaining coins from $c$ with a genuine coin. If they balance, the other coin is the counterfeit; otherwise, we can determine whether the counterfeit coin is lighter or heavier than the genuine coin.

If the side with the genuine coin is heavier in the second weighing, then the counterfeit coin on that side is heavier, and the coins on the other side are lighter than the genuine coin, if not genuine. Thus, weighing the two remaining suspicious coins will again reveal which coin is counterfeit and whether it is heavier or lighter than the genuine coin.

Three weighings are therefore always sufficient to find the counterfeit coin. Two weighings are not sufficient because with the first weighing, we can only divide the coins into at most three groups, so it is always possible that a group of at least four coins will only be known to contain the counterfeit coin, which could be lighter than the genuine coins. In this case, even with the help of genuine coins, the counterfeit coin cannot be found, only by chance, because we can only distinguish between three groups, and at least two coins will remain in one of these groups.