A Baron, a Count, a Duke and an Earl met at a jousting tournament. In the first round, two met in the first joust, and the other two met in the second joust; the two winners from the first round met at the second round for the final joust. After the jousting, they declared:

Baron: I beat the Earl.

Count: I faced both the Baron and the Duke.

Duke: I didn't make it past the first round.

Earl: At the first round, I lost to the Duke.

I knew how many were knights, and how many were liars (though not who was what) but that wasn't enough to know what jousts there had been.

However, I happened to know that a certain joust had taken place (though I didn't know who won and if it had been in the first or the second round) and that allowed me to know every result.

Can you deduce this?

(In reply to re: Ignorance is key - help! by owl)

I don't think that the argument for why the first statement is true is simple or obvious, and I needed to work fairly hard to convince myself of it before I posted. Let me see if I can elaborate.

The problem says that just knowing how many knights and liars there were wasn't enough to determine what jousts took place.

The problem also says that with one tiny extra bit of info -- namely that a specific joust happened but NOT who won it -- the speaker then was able to determine the complete results of the tournament.  The important thing here is that after knowing (a) how many liars and (b) that one joust took place but not the outcome the speaker in the puzzle knew who won the tournament (along with all the other results.)

Now, how did the speaker know who won the tournament? Item (b) doesn't say who won anything, so it's not the answer (although it helps!) No, there must be something the speaker can see in his or her analysis from (a) that actually requires a specific winner in round 2. So the next step is to figure out whose statements can force a specific winner in round 2.

Now If the Baron (B) is a knight, his statement could tell you who won round two, but only if round two was B vs E. If B is a liar, we get no help. It might be that B lost to E or that B never met E!

If the Count is a knight, then we know the Count won in round 1 and was *in* round two, but the his statement doesn't say anything about winning, just "facing" so it can't tell us anything about a winner. Likewise, if the count is a liar, we get no help -- it may be that the Count beat B and then lost to D, or lost to E in round 1, or beat E and then beat D-- we can't tell.

If the Duke is a knight, all we know is that he's a loser. And if the Duke lies, we know he won his first round, and therefore was *in* the finals (round 2) but we don't have a clue who won.

Finally, if the Earl is a knight, we know HE'S the loser. And if the Earl lies, we don't know anything. It may be that C beat E in round one or that E beat D in round 1 or that E won in round one and either won or lost in round 2 to anyone!

So, out of all of the possibilities, only one of these statements could have given the speaker a scenario where it was possible to know who won the tournament: Only if B beat E in round 2 would it be possible to know all of the results.

It may sound a bit backward -- normally you'd try to prove that something happened based on the statements the particiants made. But in this case, it's the fact that the speaker is able to know all of the results that forces this statement to be true. With anything else, the speaker would have to be lying when he said he was able to deduce all of the results.

Once you realize that the speaker's ability to solve the problem as he claimed requires that B beats E in the final round, you now can proceed to tackle the rest of the problem. Good luck, and stick with the puzzles!

Posted by Paul on 2005-08-16 04:31:05