It's Graduation Day at LKU (Liars and Knights University) where everyone enrolled is either a Liar (who always lies) or a Knight (who always tells the truth). You approach four youths who just received their diplomas, two of whom are Liars and two of whom are Knights. It is known that each of the four majored in a different one of four subjects and minored in a different one of the same four. Each makes three statements:

Roberta:
   1. Friedrich did not minor in botany.
   2. Either Gloria or myself or both took philosophy.
   3. Both girls either majored or minored in botany.

Gloria:
   1. Between them, Melrose and Roberta took all four classes.
   2. I did not minor in botany.
   3. Roberta is a Liar.

Melrose:
   1. I majored or minored in math.
   2. Friedrich is a Liar
   3. Roberta majored in history.

Friedrich:
   1. Melrose did not major or minor in math.
   2. I majored or minored in philosophy
   3. Gloria minored in math.

Which are the Liars and the Knights, and what did they major and minor in?

If we assume Frederich and Roberta are Liars, then Melrose and Gloria must be Knights - This will be the most difficult scenario to disprove becuase it gives us the least amount of information (with the two "X is a Liar" statements).  So since Melrose and Roberta took all four classes between them, only one of the two must have taken Philosophy and Gloria must have also (since if Frederick had it would make him a Knight). But that would make Roberta a Knight which is a contradiction.

So let us assume Frederich and Gloria are Liars (Now Melrose and Roberta are the Knights). That would mean Gloria did minor in botany and therefore Roberta majored in botany.  But Melrose says she majored in History.  Another contradiction.

Therefore Frederich must be a Knight!  This makes Melrose and Roberta Liars and Gloria the other Knight (Since we know Gloria minored in math, she is truthful when she says she did not minor in botany)

Knowing this it is easy to see that Frederich did minor in botany, neither girl took philosophy, meaning Melrose minored in it and Frederich majored in it.  The only remaining minor is history which was taken by Roberta, and since she and Melrose took all four subjects between them and Melrose can not have studied math, Roberta must have been the math major (although I fail to see how a Liar can graduate with a degree in Mathematics, I mean Philosophy and History, sure...).  This leaves Melrose's major as Botany and Gloria's as History.  Now that I'd like to see, a historian who can only tell the truth!

Posted by Eric on 2005-04-18 19:05:51