# Kingdom

Question:

In Josephine’s Kingdom every woman has to pass a logic exam before being allowed to marry. Every married woman knows about the fidelity of every man in the Kingdom except for her own husband, and etiquette demands that no woman should be told about the fidelity of her husband. Also, a gunshot fired in any house in the Kingdom will be heard in any other house. Queen Josephine announced that at least one unfaithful man had been discovered in the Kingdom, and that any woman knowing her husband to be unfaithful was required to shoot him at midnight following the day after she discovered his infidelity. How did the wives manage this?

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Solution:  If there is only 1 unfaithful husband, then every woman in the Kingdom knows that except for his wife, who believes that everyone is faithful. Thus, as soon as she hears from the Queen that unfaithful men exist, she knows her husband must be unfaithful, and shoots him.

If there are 2 unfaithful husbands, then both their wives believe there is only 1 unfaithful husband (the other one). Thus, they will expect that the case above will apply and that the other husband’s wife will shoot him at midnight on the next day. When no gunshot is heard, they will realize that the case above did not apply, thus there must be more than 1 unfaithful husband and (since they know that everyone else is faithful) the extra one must be their own husband.
If there are 3 unfaithful husbands, each of their wives believes there to be only 2 so they will expect that the case above will apply and both husbands will be shot on the second day. When they hear no gunshot, they will realize that the case above did not apply, thus there must be more than 2 unfaithful husbands and as before their own husband is the only candidate to be the extra one.
In general, if there are n unfaithful husbands, each of their wives will believe there to be n-1 and will expect to hear a gunshot at midnight on the n-1th day. When they do not, they know their own husband was the nth.