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Thorak

Unfaithful wives

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There is a city of wise men.
Once, a clairvoyant came, summoned all the men in the square and said: "In this town there is at least one unfaithful wife. Whoever of you finds out that his wife is unfaithful, write her name on the blackboard at the church during the night, so that her shame serves as an example for others. "
Sages thoughtfully returned to their homes. Everyone knows about all other women, whether they are faithful or not... but noone knows about his own wife. They are proud of their wisdom, so they won't tell each other.
Exactly a week later, there were names of all the unfaithful women on the blackboard.
Write comma separated numbers of names at the blackboard after the first, second... up to seventh night.
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If there is only one unfaithful wife, the husband would know


on the first night because there is a least one but he knows that
none of the other men's wives are unfaithful.
If there are two, then each man reasons thusly: If my wife is
faithful, then the husband of the unfaithful wife would know that
his wife is unfaithful (using the reasoning of the preceeding
paragraph) on the first night. Since he didn't write her name
on the first night, my wife must be unfaithful. So, on the second
night both men will write their wives name on the board and the
sequence of numbers of names on the board would be 0,2 and no more
names would be added after that.
It's an easy induction argument to show that for N unfaithful
wives the sequence of numbers of names on the board would be
N-1 nights of 0 followed by nights of the same N names on the board
thereafter.

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