Three players guess hat colors as a team — and win 75% of the time
Three players each wear a red or blue hat, seeing only the other two players' hats. Simultaneously, each one guesses their own hat's color or passes. The team wins if at least one person guesses correctly and nobody guesses wrong. What strategy gives the best odds?
Reveal the answer
Agree beforehand: if the two hats you see match each other, guess the opposite color; if they differ, pass. This wins whenever the three hats aren't all the same color — 6 of 8 equally likely combinations, a 75% win rate, far beating the 50% you'd get if one person always guessed alone. Todd Ebert posed the puzzle in his 1998 PhD thesis; the general version, for larger groups, runs on Hamming error-correcting codes.
— Todd Ebert, Hat problem — PhD thesis, UC Santa Barbara, 1998