# Married Couples

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 traveler 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?

**SOLUTION**

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.

We do not know where this puzzle originated from. If you have any information, please let us know via **email**.

There is no mention of a daily limit in the story. This is a really bad one!

Hi K P. There is no need of a daily limit in the puzzle. Can you elaborate please?

I disagree with the starting comment in the solution that women would assume the adulterer’s wife would have killed him. All the women already know about the infidelities (except theirs), and also that the other wives don’t know about their husband’s infidelity. The men know about their infidelity at least, and maybe about other men too. So when the outsider announces that there has been at least one infidelity in the village, every person would have thought “Uh huh, I already knew that”, and nothing further would have happened.

Hi Jeff. Imagine the same problem but only with 2 couples. When the stranger makes the announcement, both women will assume initially that their husbands haven’t cheated, so they will expect the other woman’s husband to be killed on the next day. However, when they see that this does not happen, they will figure out that they have been cheated on. The reasoning for 3 couples is similar, but one step more convoluted.

This problem sounds truly paradoxical, but the provided solution is in fact correct:)

Ah, but you didn’t specify whether the wives – as agents in the model – could use indirect reasoning or whether they depended on complete direct information to ensure the “killing” rule.

Yes, in these types of problems, you assume that the agents have perfect deduction skills:) Another example goes like this:

3 wise men fell asleep under a tree. A pranker passed by and painted their foreheads black. They woke up and started laughing. Each of them initially thought that the other 2 men are laughing at each other, but after a few moments, figured out that their own foreheads were painted too. How come?