Answer to riddle #9: Matriarchal Town Adultery

9. In a certain matriarchal town, the women all believe in an old prophecy that says there will come a time when a stranger will visit the town and announce whether any of the men folks are cheating on their wives. The stranger will simply say "yes" or "no", without announcing the number of men implicated or their identities. If the stranger arrives and makes his announcement, the women know that they must follow a particular rule: If on any day following the stranger's announcement a woman deduces that her husband is not faithful to her, she must kick him out into the street at 10 A.M. the next day. This action is immediately observable by every resident in the town. It is well known that each wife is already observant enough to know whether any man (except her own husband) is cheating on his wife. However, no woman can reveal that information to any other. A cheating husband is also assumed to remain silent about his infidelity.

The time comes, and a stranger arrives. He announces that there are cheating men in the town. On the morning of the 10th day following the stranger's arrival, some unfaithful men are kicked out into the street for the first time. How many of them are there?

OK the answer to this was sent to me by Eirik Lilleaas, I've sort of rephrased what he said, I hope this makes sense. I love this puzzle, pure logic. Consider the following examples...

Before reading the answer can I interest you in a clue?

One cheating husband:

If only one man were cheating then clearly his wife would work this out on the first day as she would have known that no other husbands were cheating and so the cheating husband must be hers.

Two cheating husbands:

Consider this from the point of view of Bettie one of the wives (Bettie and Anna) who are being cheated on, though all their positions are in fact the same. Bettie would be aware that Anna's husband was cheating on her and would therefore expect that she would deduce this on the first day as in the example above, because this does not happen Bettie knows that Anna is also aware of a cheating husband. Since Betty was not aware of this it must be her husband who is cheating. Anna will go through the same thought process and so two men will be thrown out on the 2nd day.

Three cheating husbands:

Charlotte will be aware Anna and Bettie's cheating husbands and expect the process to be solved as in the example above on the 2nd day, when this doesn't happen she will know that Anna and Bettie must also be aware of 2 cheating husbands and will hence chuck out her fella on the third day. Anna and Betty will think and do the same.

N cheating husbands:

Any of the wives being cheated on will be aware of N-1 cheating husbands and expect the process to be solved on the (N-1)th day, when this doesn't happen they all become aware that all the other wives that they thought were being cheated were all under the same impression and hence they must be the Nth cheated on wife. Hence on the Nth day N wives throw out N husbands.

Hence the answer, on the 10th day:

10 men are thrown out.

Great puzzle this. I love the way whilst nothing appears to be happening on each day the puzzle is being solved.

Alternative version

Every day, the prison warden takes out his 100 prisoners to play an evil game. He places them in a circle where everyone can see each other. He then places a hat on each prisoner's head. The hat is either red or white and he always gives the same hat to the same person. The prisoner cannot see the colour of his own hat. Then, on his command, all prisoners with a white hat have to step forward. If one too few or one too many steps forward, they will all be executed. If they do it right, they all go free. If they all do nothing, they all go back to their cells and the game continues the next day. If all prisoners know there is at least one person with a white hat, how many days did it take for the prisoners to get out, and how did they do it? (Note: they are not allowed to communicate in any way about the colour of their hats. For fear of execution, they don't.)

There's a certain amount we can tell from the alternative version. The significant similarities are that there must be at least one hat. There is no continuous version it has to reset each day. Doing nothing is consequence free. Perfect visibility apart from themselves.

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