Question: A certain town has 100 married couples (husband and wife only). Everyone in the town lives by the following rule:

1. If a husband cheats on his wife, the husband is executed (killed) in the following night if his wife get to know out about him (He may enjoy only as long as his wife does not get to know about it).
2. All the women in the town only gossip about the husbands of other women but no woman ever tells another woman if her husband is cheating on her. (So every woman in the town knows about all the cheating husbands in the town except her own).
3. Husbands always remain silent.

One day an outsider came to the town and some how, got to know about all the cheating and non-cheating husbands. The next day while leaving he announced to the entire town – “There is at least one cheating husband in the town”.

What will happen?

Answer: Lets consider simple cases first.

Case 1: There is only one cheating husband
99 women know who is that one cheating husband. Only one women does not know of anyone cheating. That women will then deduced that it must be her husband only who is cheating. The following night she will execute her husband.

Case 2: There are two cheating husband
98 women know that two husbands are cheating. Only two women know of one cheating husband. Those two women will expect that the other woman will kill her husband the following night (as she will deduce it as in case 1). But no one will be killed that night.

Next day those two women will deduce that there must be two cheating husband, otherwise the other women would have executed her husband. Thus those two women will deduce that their husbands are cheating and will execute them next night.

General Case: n husbands are cheating
If there are 3 husbands, it can be deduced by all women, when no one would be executed on 2nd night, and those three husbands will be killed on 3rd night Going by similar logic, all the cheating husbands will be executed on nth night.

Note: If the outsider made a false announcement, all the husbands would be executed that night as every women would think that it must be her husband only.