Heads Up: No Teamwork Required
Martin John Erickson · Mathematics Magazine · 2005
Many team games require collaborative strategies. In this Note, I present a game that paradoxically requires cooperation, but at the same time forbids communication of any kind. A team skilled in probability can compete as if they could confer (almost). In the [1], each contestant on a team tries to guess the color (blue or red) of a hat on his/her head while seeing only the hats of the other contestants. The guesses are made simultaneously, and passing is an option. The team wins if at least one person guesses and no one guesses incorrectly. Before the guessing round, the participants are allowed to discuss strategy. With best play, an n-person team (where n is one less than a power of 2) can win with the surprisingly high probability n/(n + 1). (The solution to the Hat Problem provides a novel perspective on Hamming codes, but you don't need to know anything about Hamming codes to understand this note.) I enjoyed the Hat Problem and was investigating variations of it when I thought of eliminating both the pregame strategy meeting and any information the teammates could gain from each other. The resulting game, recast in the context of coin-flipping, has some curious mathematical features.