In a small village there are 100 married couples living. Everyone in the village lives by the following two rules:
- If a husband cheats on his wife and she figures it out, the husband gets immediately killed.
- The wives gossip about all the infidelities in town, with the only exception that no woman is told whether her husband has cheated on her.
One day a traveller comes to the village and finds out that every man has cheated at least once on his wife. When he leaves, without being specific, he announces in front of everybody that at least one infidelity has occurred. What will happen in the next 100 days in the village?
Let us first see what will happen if there are N married couples in the village and K husbands have cheated, where K=1 or 2.
If K=1, then on the first day the cheating husband would get killed and nobody else will die. If K=2, then on the first day nobody will get killed. During the second day however, both women would think like this: "If my husband didn't cheat on me, then the other woman would have immediately realized that she is being cheated on and would have killed her husband on the first day. This did not happen and therefore my husband has cheated on me.". Then both men will get killed on the second day.
Now assume that if there are N couples on the island and K husbands have cheated, then all K cheaters will get killed on day K. Let us examine what will happen if there are N+1 couples on the island and L husbands have cheated.
Every woman would think like this: "If I assume that my husband didn't cheat on me, then the behavior of the remaining N couples will not be influenced by my family's presence on the island.". Therefore she has to wait and see when and how many men will get killed in the village. After L days pass however and nobody gets killed, every woman who has been cheated on will realize that her assumption is wrong and will kill her husband on the next day. Therefore if there are N+1 couples on the island, again all L cheating husbands will get killed on day L.
Applying this inductive logic consecutively for 3 couples, 4 couples, 5 couples, etc., we see that when there are 100 married couples on the island, all men will get killed on day 100.