$\textbf{Cheating Husbands}$
A remote town comprises of $100$ married couples. Everyone in the town lives by the following rule: If a husband cheats on his wife, the husband is executed at the night as soon as his wife finds out about it. All the women in the town only gossip about husbands of other women. No woman ever tells another woman if that woman's husband is cheating on her. So every woman in the town knows about all the cheating husbands in the town except her own. It can also be assumed that a husband remains silent about his infidelity. One day, the mayor of the town announces to the whole town that there is at least $1$ cheating husband in the town. What will happen afterwards?
$\textbf{Answer}$
Assuming that there are $N$ cheating husbands, then nothing will happen until the $N^{th}$ day on which all these cheating husbands will be killed.
$\textbf{Analysis}$
Let's start with the simplest case in which there is only one cheating husband. Then, his wife will immediately realizes this on the first day because she knows no other husband is cheating.
Next, assume there are two cheating husbands. Then, $98$ wives know there are two cheating husbands, but two wives only know there is one cheating husband. On day one, nothing will happen because these two wives cannot be sure whether their husbands are cheating. However, on the second day, they will both realize that there are at least two cheating husbands but they only know one. This means that their own husbands are cheating. Hence, both cheating husbands will be killed. The reason that these two wives will realize the existence of at least two cheating husbands is because if there is only one, then this man's wife will know no other cheating husband. As a result, as described in the previous case, she must come to a conclusion that her husband is cheating and kills him. The fact that nothing happens on the first day means that this is the not case.
If there are $3$ cheating husbands, then by a similar reasoning nothing will happen until the third day. And all these cheating husbands will be executed that night. Now, by induction, we can conclude that if there are $N$ cheating husbands, then they will all be executed on the $N^{th}$ day.