Two types of people live on the island, Knights and Knaves.

   Knights always tell the truth.
   Knaves always lie.

   You run into a group of six natives from the island.

   U, V, W, X, Y and Z  (They have short names on this island).

   U says: None of us is a knight.
   V says: At least three of us are knights.
   W says: At most three of us are knights.
   X says: Exactly five of us are knights.
   Y says: Exactly two of us are knights.
   Z says: Exactly one of us is a knight.

   Which are the knights, and which are the knaves?

   U is a knave, a liar.
   None of us is a knight would be true, if none of U,V,W,X,Y or Z
                                         were knights, but that 
                                         is a contradiction, because
                                         if it were TRUE, U would have
                                         to be a knight.  But if he were
                                         a knight, it would be FALSE.

    The statement is actually an impossible one for any knight to
    ever make!

    Fact #1:  U is a knave.  (Its NOT the case there are 0 knights)

    Consider the last statement, by Z.

    Z says: Exactly one of us is a knight.
    
    Suppose that this statement is TRUE and Z is a knight.

    Oops!  That would make W's statement true too.

           W says: At most three of us are knights.

           But if W's statement is true, W is a knight too,
           along with Z, so Z's statement cannot possibly be
           true, because it leads to a CONTRADICTION, or 
           IMPOSSIBILITY or ABSURDITY.

    Fact #2:  Z is a knave.   (Its NOT the case there is ONE knight).

    Since we have that U is a knave and Z is a knave, we now
       have a total of 2 knaves.
 
    Since we only have V, W, X and Y left, we now know that X is a knave!

    X says that exactly five of us are knights, and at this point we know
    that there are FOUR OR FEWER knights.

    Therefore, X is a liar, the knave, indeed.

    Fact #3:  X is a knave.   (Its NOT the case there are FIVE knights).
     
    What do we have left now?

    U, X and Z are knaves, for a total of THREE knaves.

      V says: At least three of us are knights.
      W says: At most three of us are knights.

      Y says: Exactly two of us are knights.

    Consider Y's statement, that there are EXACTLY TWO knights.

    **** "There are EXACTLY TWO knights", has EXACTLY TWO cases. ****

    Case #1:  Suppose its TRUE.

    Y is a knight.
    W's statement that AT MOST THREE OF US ARE KNIGHTS, is true.

    V's statement that AT LEAST THREE OF US ARE KNIGHTS, if false.

    Y is knight.   W is knight.  V is knave.  This is possible....
                                                      --------

    Case #2:  Suppose its FALSE.

    Y is a knave.  There are NOT exactly TWO knights.

    W's statement that AT MOST THREE OF US ARE KNIGHTS, is true.

    V's statement that AT LEAST THREE OF US ARE KNIGHTS, is false.    

    W is a knight, but Y and V are both knaves.

    But from above, we already have that U, X, and Z are knaves.
    So U, V, W, X and Z are knaves and W is the one knight.

    But that is impossible, because ONE KNIGHT would make Z's 
    statement TRUE, and that is impossible.  It was ruled out above.

    Conclusion:
                W and Y and knights.           U, V, X, and Z are knaves.