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:

1. An Odd shakes with an Odd. In this case they both become Evens; the total number of Odds decreases by 2.
2. An Even shakes with an Even. Both become Odds, and the number of Odds increases by 2.
3. An Even shakes with an Odd. The Even becomes an Odd, and the Odd becomes an Even, so the total number of Odds remains unchanged.

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.)”