The night the husbands were thrown out

Question

A town has 100 married couples. Some husbands are secretly unfaithful. Gossip is perfect in one direction: every wife knows exactly which husbands other than her own are unfaithful, but nobody ever tells a wife about her own husband. The law says that a wife who deduces her husband is unfaithful must throw him out at midnight that same night.

One morning the mayor announces publicly: "At least one husband in this town is unfaithful."

For nine nights nothing happens. On the tenth night, several husbands are thrown out at once.

How many husbands were unfaithful?

Accepted Answer

Exactly 10 husbands were unfaithful, and all 10 were thrown out on the tenth night. One cheater. His wife knows of no other cheaters. The mayor says there is at least one. So it is her husband; she throws him out on night 1. Two cheaters. Each of their wives knows of exactly one other cheater and thinks: "Maybe he is the only one." If so, that man's wife would throw him out on night 1. Night 1 passes with nothing happening. Each wife now knows there must be a second cheater, and the only candidate she cannot see is her own husband. Both act on night 2. k cheaters. Each of their wives sees k m…

The night the husbands were thrown out

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