Logic Puzzles


 

15. The recent expedition to the lost city of Atlantis discovered scrolls attributed to the great poet, scholar, philosopher Josephine. They number eight in all, and here is the first. The kingdom of Mamajorca, was ruled by queen Henrietta I. In Mamajorca women have to pass an extensive logic exam before they are allowed to get married. Queens do not have to take this exam. All the women in Mamajorca are loyal to their queen and do whatever she tells them to. The queens of Mamajorca are truthful. All shots fired in Mamajorca can be heard in every house. All above facts are known to be common knowledge. Henrietta was worried about the infidelity of the married men in Mamajorca. She summoned all the wives to the town square,
and made the following announcement. "There is at least one unfaithful husband in Mamajorca. All wives know which husbands  are unfaithful, but have no knowledge about the fidelity of their own husband. You are forbidden to discuss your husband's faithfulness with any other woman. If you discover that your husband is unfaithful, you must shoot him at precisely midnight
of the day you find that out." Thirty-nine silent nights followed the queen's announcement. On the fortieth night, shots were heard. How many husbands were shot on that fateful night?

Solution: If there are n unfaithful husbands (UHs), every wife of an UH knows of n-1 UH's while every wife of a faithful husband knows of n UHs. [this because everyone has perfect information about everything except the fidelity of their own husband]. Now we do a simple induction: Assume that there is only one UH. Then all the wives but one know that there is just one UH, but the  wife of the UH thinks that everyone is faithful. Upon hearing that "there is at least one UH", the wife realizes that the  only husband it can be is her own, and so shoots him. Now, imagine that there are just two UH's. Each wife of an UH assumes that the situation is "only one UH in town" and so waits to hear the other wife (she knows who it is, of course) shoot her husband on the first night. When no one is shot, that can only be because her OWN husband was a second UH. The wife of the second UH makes the same deduction when no shot is fired the first night (she was waiting, and expecting the other to shoot, too). So they both figure it out after the first night, and shoot their husbands the second night. It is easy to tidy up the induction to show that the n UHs will all be shot just on the n'th midnight.

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