Talk:Knights and Knaves

From Wikipedia, the free encyclopedia

I'm not certain that Smullyan first did these, but he certainly does many of them. To my mind, the ancestor of these puzzles is the old story about the two guards on two doors, one with treasuer, one with a tiger, and one guard lies, one tells the truth, and you only have one question to ask. Does this old one have a specific name? Where does it originate? (it has a 1001 nights feel?) -- Tarquin 21:48 19 Jul 2003 (UTC)

Regarding Question 2 - Without using any formal boolean algebra, a different solution came to me immediately - it appears to me that it could also be that both Bill and John are knaves (in this case, both Bill and John's statements are false, which is consistent with them both being knaves). Any problem with that solution? It seems a much simpler reasoning than the solution presented in section 1.5... Zoopee 09:09, 26 August 2006 (UTC)

I believe that's correct, they're both knaves, since they can't both be knights because of Bill's statement, they can't be a knave and a knight because then John's statement wouldn't be false, and they can't be a knight and a knave because then Bill's statement would be true. Which leaves us with two knaves. The solution to the puzzle isn't given in the article though, I think it was just meant as an example.--BigCow 21:18, 21 November 2006 (UTC)
yes, both of you are right, they are both knaves. I wrote the solution in the italian version but I can't write it in good english --Arirossa 20:55, 19 January 2007 (UTC)
On the other hand, if Bill is a knight and John is a knave, it works: Bill's statement that they are different is truthful, and, seeing as the requirements for John's statement to have meaning are unfulfilled, it could well be that he is lying and Bill's position has no bearing on John. The 2-knaves answer obviously works as well, so this has more than one answer.--68.100.78.205 20:06, 24 March 2007 (UTC)
I don't think that is correct. It doesn't work if Bill is a knight and John is a knave. The reason being, if John is a knave then the statement "If Bill is a knave, then John is a knight." is false. And when negated (assuming it is a conditional, not biconditional) it turns into "Bill is a knave and John is a knave." Thus if what John says is false, Bill can not be a knight. This is my math: (if not p, then q) = (p or q), so not(if not p, then q) = not(p or q) = not p and not q. --[I'm not a user] time: 11:52, September 23, 2007 —Preceding unsigned comment added by 24.12.147.8 (talk) 04:54, 24 September 2007 (UTC)
I have now edited the article to make it clearer why the situation doesnt work.Yarilo2 (talk) 01:28, 10 March 2008 (UTC)
I think Question 2 is unsolvable: Suppose Bill is a knave, then John is effectively saying that he is a knave. This is impossible, therefore Bill is a knight. This makes John's statement automatically true (P->Q is true if P is false), hence John is also a knight. But then John and Bill are not different, in contradiction to what Bill (who is a knight) says. Without Bill's statement, the question would be solvable (both are knights). 212.61.144.123 (talk) 09:09, 3 June 2008 (UTC)

Answer to 3, in case someone can put it in boolean: John answers yes to the first question (if he answered no, he would have to be a knight and the logician would figure it out immediately) and yes to the second (if he answered no, the answer would remain unclear). He is a knave and Bill is a knight; allowing him to say no yes both times (since John and Bill are different, both yeses are untrue).--68.100.78.205 20:15, 24 March 2007 (UTC)

For 4: The answer is "Are you the sort of person who would say that Someplaceorother is to the left?" Nobody would claim to be a knave, so if they say yes, it is to the left, and if no, it is to the right. The second problem posed has nothing to do with the question: "Does 2+2=4?" would do, or any obvious equivalent. --68.100.78.205 20:19, 24 March 2007 (UTC)

I think the question to be to one of them could be: Would that other guy say that Someplace is to the left?
If you are asking the Knight, and Someplace is actually to the right, the Knight will answer YES. If you are asking the Knave and Someplace is actually to the right, the Knave will also say YES because we know that if the Knave were to speak the truth, he would say that the answer from another truth-teller would be NO.
If you are asking the Knight, and Someplace is actually to the left, the Knight will answer NO because he knows that the Knave would lie about the correct direction. If you are asking the Knave and Someplace is actually to the left, the Knave will also say NO because we know that if the Knave were to speak the truth, he would say that the answer from another truth-teller would be YES. But he will lie and say NO.
Having received YES, the traveler knows that Someplace is actually to the right. If the traveler received NO, the correct direction would be to the left.
I'm reasonably sure that one of the premises of this quiz has to be that you cannot ask a question to which you already know the answer. So asking if 2+2=4 would fall out as improper. I think that the second question would have to be: Would that other guy say you are a Knight?
If you are asking the Knight, he will answer NO since he knows the Knave will lie about him. If you are asking the Knave, he will respond YES because we know that if the Knave were to momentarily speak the truth, he would say that the answer from another truth-teller would be NO. But he will lie and say YES.
As a result if you hear NO, you are speaking to the Knight. If you hear YES, you have asked the Knave instead.
Implicit in this is the assumption that the Knights and Knaves all know who each of them are. I think this article should state that more clearly as a premise. Move over, I think that it should be made clear that you cannot ask a question to which you already know the answer. That would defeat the whole point of the puzzle. Dawgknot 21:42, 24 August 2007 (UTC)
In question 4, you can disregard one person completely to get the right path. "Does this path I'm pointing to lead to freedom being true, the same as you being a Knight?". In terms of Boolean equality, this holds true as follows: Let Q be the question, A and B being the two people, and L being the truth of the selected path leading to freedom. And hence we formulate the problem in terms of Boolean Equality as follows: Q ≡ A ≡ L ∧ A ≡ ￢B. From here, we already have a solution. Q ≡ (A ≡ L), which is the question stated above, where you take the path if the answer is Yes, or take the other path if the answer is No (That is, the truth value of the question Q). B can be completely disregarded, as it is unimportant in this scenario.
Let's do a case analysis too:
Case 1: Path is correct, and A is a Knight: A says YES (True ≡ True is True. Knight tells the truth and will agree).
Case 2: Path is wrong, and A is a Knight: A says NO. (False ≡ True is False. Knight tells the truth and will disagree).
Case 3: Path is correct, and A is a Knave: A says YES. (True ≡ False is False; Knave lies and will agree).
Case 4: Path is wrong, and A is a Knave: A says NO. (False ≡ False is True; Knave lies and will disagree).
And there you have it.
Regarding the next part of Question 4, I believe that stepping on the foot of the knight is cheating; you're just supposed to ask a question, and not do anything else. I personally would ask a question based on the premise that Q ≡ A ≡ (A ≡ True). That is, Question posed to A whether A is a Knight. Simplify this: Q ≡ (A ≡ A) ≡ True (Associativity of equivalence), then Q ≡ True ≡ True, and finally Q ≡ True. So just ask any question that the answer is already known, i.e. 1 + 1?, or "Am I asking you a question?, or "Is True 'True'?" (ha ha), or ask a logic puzzle :). 203.188.235.14 (talk) 06:28, 8 March 2008 (UTC)

Contents

[edit] Solution to Question 1

Here is the text in the article after establishing that John is knave.

Since knaves lie, and one statement is true, the other statement must be false

Wouldn't it be clearer, given the construction of the Question, to say:

Since knaves lie, then both statements cannot be true. Therefore, the other statement must be false

Also, is it an unspoken assumption that the knights and the knaves know among themselves which is which? That is to say that John and Bill know the truth about each other? If so, perhaps that should be premised. Dawgknot 21:00, 24 August 2007 (UTC)

[edit] More popular culture references

I think there's an episode of Samurai Jack that features a version of this puzzle, but I can't recall which episode. B7T 20:11, 2 December 2007 (UTC)

[edit] Boolean Equality

Has anyone ever heard of this here? It's a very good read in regards to the problem of the Knights and Knaves. 203.188.235.14 (talk) 14:41, 18 January 2008 (UTC)

[edit] Intro

An early example of this type of puzzle involves three inhabitants referred to as A, B and C. The visitor asks A what type he is, but does not hear A's answer. B then says "A said that he is a knave" and C says "Don't believe B: he is lying!" To solve the puzzle, note that no inhabitant can say that he is a knave. Therefore B's statement must be untrue, so he is a knave, making C's statement true, so he is a knight and A is a knight.

I don't know where this example comes from, but I don't think it's true. We can't make any conclusions about A in this puzzle, the statements of the other two work whether A is a knight or a knave.

99.241.128.44 (talk) 06:44, 11 February 2008 (UTC)Alex

[edit] Question 1: fallacy?

Question one is really strange.

Look at this statement: "If John was a knight, he would not be able to say that he was a knave since he would be lying. Therefore the statement 'John is a knave' must be true."

Sentence one is right, but sentence one does not imply sentence two at all. If John was a knave, he would not be able to say that was a knave either, because he would be telling the truth. This contradicts with the fact that knaves always lie.

Also, although I'm not an expert on Boolean algebra, could an expert verify if the Boolean algebra performed is correct?

--Freiddie 12:22, 26 May 2008 (UTC)

It is very subtle: There are not two statements "John is a knave" and "Bill is a knave". In that case you would indeed get a contradiction. There is a single statement: "John is a knave AND Bill is a knave". John as a knight could not say that, because the statement would be untrue (since John would not be a knave), hence John must be a knave. Then the statement "John is a knave and Bill is a knave" must be untrue. Since John is a knave, then Bill cannot also be a knave (then the statement would be true), so Bill is a knight, and the statement is untrue (because Bill is not a knave, although John is). 212.61.144.123 (talk) 09:26, 3 June 2008 (UTC)
Oh I see now. The "AND" operator is also being said, but it's not assumed. Okay, I get it now. Thank you. --Freiddie 00:59, 5 June 2008 (UTC) —Preceding unsigned comment added by Freiddie (talk • contribs)