A certain strange casino offers only one game. The casino posts a positive integer *n* on the wall, and the customer flips a fair coin repeatedly until it falls tails. If he has tossed *n* – 1 times, he pays the house 8^{n – 1} dollars; if he’s tossed *n* + 1 times, the house pays him 8^{n} dollars; and in all other cases the payoff is zero.

The probability of tossing the coin exactly *n* times is 1/2^{n}, so the customer’s expected winnings are 8^{n}/2^{n + 1} – 8^{n – 1}/2^{n – 1} = 4^{n – 1} for *n* > 1, and 2 for *n* = 1. So his expected gain is positive.

But suppose it turns out that the casino arrived at the number *n* by tossing the same fair coin and counting the tosses, up to and including the first tails. This presents a puzzle: “You and the house are behaving in a completely symmetric manner,” writes David Gale in *Tracking the Automatic ANT* (1998). “Each of you tosses the coin, and if the number of tosses happens to be the consecutive integers *n* and *n* + 1, then the *n*-tosser pays the (*n* + 1)-tosser 8^{n} dollars. But we have just seen that the game is to your advantage as measured by expectation no matter what number the house announces. How can there be this asymmetry in a completely symmetric game?”