Black and white hats (1)
There are so many problems with guys sitting in a circle, each wearing a white or a black hat, that one can write a book about them. Here is one cute puzzle of this kind I've recently heard:
20 persons sit in a circle, each of them has a hat. Each person can see the hats worn by the others, but cannot see his own hat. At some moment in time, all the persons must simultaneously guess the color of the hat they have. The team loses if at least 11 persons give the wrong answer, wins if at least 11 persons give the right answer and continues the game if exactly 10 persons guess the right color. Is there a strategy to ensure that the team will never lose?