[math-fun] Hat Game

Dear funsters, The Hat Game problem below came to light over 10 years ago and the best strategy is not yet known for arbitrary N. It is known when N is one less than a power of 2. Can anyone tell me the best strategy for N=5 and for N=6? Happy puzzling, Dick THE HAT GAME N PLAYERS ENTER A ROOM AND A BLACK OR WHITE HAT IS PLACED ON EACH PERSON'S HEAD AS DETERMINED BY A FAIR COIN TOSS. EACH SEES THE HATS ON THE OTHERS BUT NOT HIS OWN. AT A SIGNAL EACH PLAYER MUST SIMULTANEOUSLY EITHER ANNOUNCE THE COLOR OF HIS HAT OR PASS. NO COMMUNICATION IS ALLOWED EXCEPT FOR AN INITIAL STRATEGY SESSION. THE GROUP SHARES A LARGE PRIZE IF AT LEAST ONE GUESSES CORRECTLY AND NO PLAYER GUESSES INCORRECTLY. CAN THEY DO ANY BETTER THAN A 50% CHANCE?