Esther is engaged. Her fiancé is either Arthur, Barton, Claude, or Dexter.
- Each of the four men and Esther is a knight or a liar.
- Arthur says: "Exactly one of us four men is a knight."
- Barton says: "Exactly one of us four men is a liar."
- Claude says: "Arthur or Barton is Esther’s fiancé."
- Esther says: "My fiancé and I are either
both knights, or both of us are liars."
Who is Esther's fiancé?
Suppose Esther is a knight. Then, her fiance is of the same type, so he is a knight. Suppose Esther is a liar. Then, her fiance is of a different type, so he is a knight. Therefore, Esther's fiance is a knight. Now, suppose Arthur is a knight. Then, he is the only knight, so he is Esther's fiance. However, that makes Claude a knight. Therefore, Arthur is not a knight. Since he is not a knight, there cannot be only one knight among the men. Since the fiance is a knight, there must be at least two knights. Suppose Barton is a liar. Then, Arthur and Barton are both liars, so the remaining two are both knights. Then, Claude is a knight, so either Arthur or Barton is the fiance. However, they are both liars and the fiance is a knight. Therefore, Barton is a knight. That means that there is only one liar, so Arthur is the only liar. Since Claude is a knight, Esther's fiance is either Arthur or Barton. Since the fiance is a knight, the fiance is Barton.
Posted by Math Man
on 2013-03-25 17:27:35