Every integer greater than 8 has at least two letters in common with each of its neighbours (source).
When the magician leaves the room, the trickees lay out eight coins in a row deciding which side is turned up according to their whim. They also think of a number between 1 and 8 inclusive. The magician’s assistant then flips exactly one of the coins, before inviting the magician back in. The magician looks at the coins and guesses the number that the trickees thought of.
“Last semester I offered my students $1,000,000 dollars. They
turned me down. This was lucky, despite the money and glamour of
academic mathematics, I do not have a million dollars. The game was
simple. The class of 100 each had to write a number. The highest number
won. Of course there was a catch, the prize was $1,000,000 divided by
the winning number. The best outcome for the students as a whole would
come if everyone wrote 1, $10,000 is not a bad return for a lecture. Of
course if everyone is writing 1, the person who writes 2 wins and makes
far more for themselves. What happened?” (Source)