DEPARTMENT OF MATHEMATICS
       ALGEBRA AND COMBINATORICS SEMINAR

    MONDAY 5 JULY 2004, 11am in Room 67-343


                   Hats

               W. D. Wallis

     Southern Illinois University, Carbondale

Abstract
In a game show, a team of  n  players competes for a
shared prize of  $1,000,000.  Each contestant enters
the studio blindfolded, and a hat is placed on his/her
head.  The hats are either black or red. When
they are all hatted, the blindfolds are removed.  A
contestant cannot see his/her own hat, but can see
all the others.  No communication is allowed.  Each
contestant has to guess the color of his/her own hat
(the contestant is also allowed to pass).  They write
down their answers -- either red or black or pass
-- independently and simultaneously, so that none has
any idea of the other players' responses.  If 
there is even one wrong answer, they lose.  If they
all pass, they lose.  To win the money at least one
player must guess the correct color, and no one gets
it wrong (although some of them, all but one in the
extreme case, may pass).

The players are told the rules.  They are also told
that the allocation of hat colors is independent and
random, with each player having 50% chance of red
(for example, a fair coin is tossed for each
player).  They can then decide on a strategy.  But
remember, once the game starts there is no
communication, and no player knows any other player's
response before making his/her own.

What is the team's best strategy?

We shall discuss this problem and its relationship to
coding theory.


All welcome.