Prove that the number of people who shake hands an odd number of times at the opera next Thursday will be even.
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Call a person an Odd if he has shaken hands an odd number of times (at any given moment). An Even has shaken hands an even number of times.
At the start, the number of Odds is zero, an even number. As the evening progresses, three types of handshake take place:
Because the number of Odds can change only by an even number, the number of people who have shaken hands an odd number of times remains even — not only at the end of the evening, but at every moment throughout!
A Norwegian reader adds: “It might possibly be of some minor interest that in Norway, Odd and Even are both widespread men’s names. Fittingly, as of the most recently updated name statistics (2009), there are 22821 Norwegian men with Odd as their first first name, and 5846 with Even as their first first name. (And 29 are called Odd Even.)”