Each of the three girls Francine, Gertrude and Harriet are either knights who always tell the truth or liars who always tell a lie.
Francine says, "Either I belong or Gertrude belongs to a different type from the other two."
Whose type can you deduce with absolute certainty?
Suppose Francine is a knight. Then, either she or Gertrude is a different type from the other two. If Francine is different, then the other two are liars. If Gertrude is different, then the other two are knights. In both cases, Gertrude is a liar.
Suppose Francine is a liar. Then, neither she nor Gertrude is a unique type. Suppose Gertrude is a knight. Then, Harriet cannot be a knight because Francine would be the only liar. However, Harriet cannot be a liar because Gertrude would be the only knight. This is a contradiction, so Gertrude is a liar.
In all cases, Gertrude is a liar. Therefore, Gertrude is the one that can be deduced. Gertrude is a liar.

Posted by Math Man
on 20111224 13:50:04 