A knight remembers a banquet that he had:

"Oh, that was a really nice banquet; all the liars and the knights of the kingdom were there, even the village fool. We all ate and drank and at the end of it the king asked each one of us to make a statement about someone else, I mean, to say if he is a liar or a knight. finaly, one statement was said about each one of us, and every one made the same statement except for the village fool who made a different statement. By the way, can you tell me how many people were in the banquet?"

"No", you say, "but I can tell you something else about the NUMBER of people that were in the banquet."

What can you tell? What statement did each one of the participants make?

(In reply to Another way by AvalonXQ)

This is a good point. I will try to explain it to go along with my previous comments.

In the "solution with limitations", we can conclude that they can't have said "he's a knight" because that leads to a contradiction. In the "solution without limitations", that is no longer an issue, because the chain is broken with the statement the fool says.

The cycles in this problem are similar to those in "Love me for a reason" (http://perplexus.info/show.php?pid=516) in that you can have separate cycles. (using the idea of putting them in a line such that each person makes a statement about the person ahead of them in line) if someone (person X) instead makes a statement about someone who already spoke (not ahead of person X in line), then those people (from person X back) form a separate cycle from all the people ahead of him in line. (Nobody ahead of person X in line can talk about anyone before him, because a statement has already been made about each of them.)

The reason there doesn't need to be an equal number of liars and knights then, is because all knights/liars can say "he is a knight" -- the statement can be made by a knight talking about a knight, or a liar talking about a liar.

So put the all the liars in one (or more) cycle and all the knights in another cycle. Then put the fool in one of the liars' cycles (if his statement or type doesn't matter) and you have any number of liars and knights in cycles, all of which are consistent.

It says in the problem that something could be told about the number of people attending, so the fool must have some limitations on what he is and says to make this possible.

Posted by Gamer on 2007-02-11 15:18:15