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The Master of Trinity College and his wife invite n fellows and their wives to a party. During the party a fellow may shake hands with anyone except himself and his wife (assume all fellows are male, and that they'll only shake hands with someone once).
At the end of the party, the Master asks everyone else in the room how many people they shook hands with, and receives 2n+1 different answers.
i) Show that the Master's wife didn't shake hands with the most people.
ii) How many hands does the Master shake?
Source: James Michael O'Driscoll, Ask NRICH web message board.
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