Could You Solve This $1 Million Hat Trick?

A simple new puzzle has been tormenting mathematicians and computer scientists since Dr. Todd Ebert, a computer science professor at the University of California-Irvine, introduced it in his Ph.D. thesis in 1998.

Taken up more recently by Peter Winkler of Bell Labs and other puzzle mavens throughout the country, it has become the subject of discussion at high-tech companies, science chat rooms, and university math and computer science departments, in part because of its applications to so-called error-correcting codes.

The Puzzle

Here's the situation. Three people enter a room sequentially and a red or a blue hat is placed on each of their heads depending upon whether a coin lands heads or tails.

Once in the room, they can see the hat color of each of the other two people but not their own hat color. They can't communicate with each other in any way, but each has the option of guessing the color of his or her own hat. If at least one person guesses right and no one guesses wrong, they'll each win a million dollars. If no one guesses correctly or at least one person guesses wrong, they win nothing.

The three people are allowed to confer about a possible strategy before entering the room, however. They may decide, for example, that only one designated person will guess his own hat color and the other two will remain silent, a strategy that will result in a 50 percent chance of winning the money. Can they come up with a strategy that works more frequently?

Most observers think that this is impossible because the hat colors are independent of each other and none of the three people can learn anything about his or her hat color by looking at the hat colors of the others. Any guess is as likely to be wrong as right.

Is there a strategy the group can follow that results in its winning the money more than 50 percent of the time? The solution and a discussion are below, but you might want to think about the problem before reading on.

The Solution

There is, in fact, a strategy that enables the group to win 75 percent of the time. It requires each one of the three to inspect the hat colors of the other two and, if the colors are the same, then to guess his or her own hat to be the opposite color. When any one of the three sees that the hat colors of the other two differ, then he or she must remain silent and not make a guess.

A listing of all possible hat colors for the three helps to see why this strategy works. The eight possibilities for the hat colors of the three people are RRR, RRB, RBR, BRR, BBR, BRB, RBB, and BBB, the first entry indicating all three wearing red hats; the second indicating the first two wearing red and the third one wearing blue; the third indicating the first and third wearing red and second wearing blue; and so on.

In six of the eight possibilities (in bold type), exactly two of the three people have the same color hat. In these six cases, both of these people would remain silent (why?), but the remaining person, seeing the same hat color on the other two, would guess the opposite color for his or her own hat and be right. In two of the eight possibilities (in italics), all three have the same color hat and so each of the three would guess that their hats were the opposite color and all three of them would be wrong.

So when they're wrong, they're very wrong, but when they're right, they're just right enough.

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