A Coin Trick

Han tosses 12 coins onto a table top. He closes his eyes and instructs John to turn over as many coins as he likes. John can, if he wishes, turn over the same coin every time or any number of times, but there is one proviso: Every time a coin is turned John must say out loud the word “flip.”

When finished, John covers one coin with his hand and tells Han it is okay to open his eyes. Han then swiftly, and correctly, announces the state of the coin under John's hand, whether it is heads up or tails up. Han is able to do this correctly every time the game is played, even if a different number of coins is used. What is Han's trick?

Comment. When performing this trick in front of a large group, consider using chips colored black on one side and white on the other rather than coins for better visibility.

Let's Shake Hands

With an odd number of people in the room, ask everyone to shake hands an odd number of times. No person need shake everyone's hand. In fact, each person could just shake hands with the same small selection of people multiple times. All that is required is that every person be involved in an odd number of handshakes. Noting that it is impolite to refuse a handshake when offered (and that shaking hands with yourself is considered invalid), what curious predicament do folks find themselves in when they attempt this experiment?