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?

We are given two possibilities, that all said "he's a knight" (and the fool said "He's a liar") or that all said "he's a liar" (and the fool said "he's a knight")

Since each person referred to another once, and was referred to once, there is a "chain" of comments which includes the fool. In that chain, we can look at what statements were said.

If everyone said "he's a knight", then whoever they were talking about would have been the same type as them. Thus, everyone in the fool's chain would be of the same type, but then, the fool would have to say "he's a knight" as well, leading to a contradiction.

So everyone must have said "he's a liar", and the fool said "he's a knight" -- this implies the who said it was different type than who he was referring to. We know that the fool must have been the same type as the person he referred to, and so if the fool was excluded (the person who refers to the fool, instead refers to the person the fool talked about) it would still work. (This would be the situation in any chains in which the fool was not included.) Thus, there must be the same number of both, and an even number altogether, or odd if the fool was included.

Posted by Gamer on 2007-02-10 13:46:37