First, if two people had blue hats, not everyone's hand would have been raised. Next, if player 1 had seen a blue hat on player 2 & a red hat on player 3, then player 1 would have known immediately that his own hat must be red. Thus any player who sees a blue hat can guess at once. Finally, the winner realizes that since no one guesses at once, there must be no blue hats, so every hat must be red.