Alex, Bert, and Carl are taking a break from being main subjects in these logic puzzles, so Dave and Eddy decided to comment on the rarely seen Fred, Gary, and Hank.

Dave and Eddy are both knaves and each one makes four of the eight statements below. The statements are in order, but whether Dave or Eddy made any given statement is not known. Without knowing which statements are Dave's and which are Eddy's, can you determine the types of Fred, Gary, and Hank?

1. Fred is a liar.
2. Gary is a knave.
3. Hank is a knight.
4. Fred and Gary are the same type.
5. Gary and Hank are different types.
6. Fred is a knight.
7. Hank is a knave.
8. Gary is a liar.

Know thy Knaves

Solution: Fred and Hank are Knights, Gary is a Liar.

Explanation: There are 27 possible ways to assign the three types to three people.  For each of these, evaluate the eight statements given.  Since Dave and Eddy are both knaves, they must alternate TFTF or FTFT, each making four statements.  Of the 27, there are only 6 ways that give four true statements and four false statements.

Of those six possibilities, five are excluded because the eight statements (given "in order") cannot be distributed to Dave and Eddy as TFTF or FTFT. This leaves the remaining option:

The statements are FFTFTTFT.  The first, third, four, and fifth statements are by one of them and are FTFT.  The second, sixth, seventh, eighth are by the other of them and are also FTFT.  We have no way to know which is Dave and which Eddy. 

I'm not sure what to do next, so I'll submit, and see what happens.

Posted by badger on 2007-10-10 16:42:48