a logic puzzle

Nick Wedd wrote: ) In message , ehsjr )>That's three answers, which violates the rule by assuming )>that more than one person will answer a question addressed )>to a single person: )>"You are permitted 2 )>questions, each question addressed to a single (not )>both the same) person, and must determine their )>identities." ) ) It doesn't violate that rule. You ask the question once, to one person, ) who gives one of the three possible answers listed above. Then you ask ) it to one other person. Like this:

Exactly. However:

) You ask A: "What is the largest currently known prime number?" ) A says "12". ) You ask B: "What is the largest currently known prime number?" ) B says "no". ) ) So A is the knave, B is the randomiser, and C is the knight.

Almost. You see, the knave and knight are only required to answer yes/no questions. So you need a question that *is* a yas/no question, but that cannot be answered truthfully or lyingly with yes or no.

SaSW, Willem

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#EOT

PeterD wrote: ) On Thu, 26 Mar 2009 18:58:09 +1100, Sylvia Else ) wrote: )>And yet that's not quite true either: )>

)>Q1 to P1 - If you are the knave, then is P2 the randomizer, else if you )>are the knight, then is P3 the randomizer, else do you like eggs? )>

)>If P1 is the knave, )> and P2 is the randomizer, then answers no -> P3 not randomizer. ) ) Nope: if P2 is the randomizer, then he'd answer either yes or no ) randomly. That is what the randomizer does, you can't trust his ) answer, half the time it is going to be wrong.

If P2 is the randomizer then P3 is not the randomizer.

Duh.

SaSW, Willem

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Disclaimer: I am in no way responsible for any of the statements
            made in the above text. For all I know I might be
            drugged or something..
            No I\'m not paranoid. You all think I\'m paranoid, don\'t you !
#EOT

Nick Wedd wrote: ) I wonder what clause you mean. The version of the puzzle which appeared ) on my news server says ) "You know the puzzles involving Dichotomy Island, inhabited ) by knights and knaves, who will answer any yes/no question; ) knights are honest, knaves are compulsive liars." ) It does not forbid asking other questions. It even specifies that ) "randomizers .. answer any question yes or no, 50-50."

However, it allows the knights and knaves to not answer a question that is not a yes/no question. And assuming that the three will, within the rules, make it as difficult as possible for you, this means that you can only use a non yes/no question to identify if the questionee is or is not the randomizer.

SaSW, Willem

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Disclaimer: I am in no way responsible for any of the statements
            made in the above text. For all I know I might be
            drugged or something..
            No I\'m not paranoid. You all think I\'m paranoid, don\'t you !
#EOT

Three island inhabitants have six possible configurations:

1 Knight, Knave, Randomizer 2 Knave, Randomizer, Knight 3 Randomizer, Knight, Knave 4 Knight, Randomizer, Knave 5 Randomizer, Knave, Knight 6 Knave, Knight, Randomizer

Two qustions limited to yes/know give four possible sequences of answers:

1 yes, no 2 no, yes 3 yes, yes 4 no, no

This is not enough to tell six combinations apart.

Allowing an IDNCA ("I don't know/cannot answer") response and allowing two questions gives us nine possible sequences of answers:

1 Yes, No 2 No, IDNCA 3 IDNCA, Yes 4 Yes, IDNCA 5 IDNCA, No 6 No, Yes 7 Yes, Yes 8 No, No 9 IDNCA, IDNCA

That's enough combinations (which is necessary, but is it sufficient?). Thus at least one island inhabitant must be allowed an "I don't know/ cannot answer") response.

Do we allow the Knight the "I don't know/cannot answer" response? If not, he isn't really a Knight; there are questions that he must answer with a lie (yes and no are both lies if the correct answer is "I cannot answer.") Thus the Knight must be allowed the "I don't know/cannot answer" response.

Do we allow the randomizer the "I don't know/cannot answer" response? (Presumably he would say that one time out of three.) If so, answers from the randomizer give zero additional information. However, the "the randomizers [...] answer any question yes or no, 50-50" claise disallows the randomizer the "I don't know/cannot answer" response. Thus the Ranomizer must not be allowed the "I don't know/cannot answer" response, and any "I don't know/cannot answer" response is also an "I am not the randomizer" response.

If we allow the Knave the "I don't know/cannot answer" response, do we allow the knave to tell lies about whether he doesn't know/cannot answer? If not, he isn't really a knave; there are questions that he must answer truthfully. Thus, if the knave is allowed the "I don't know/cannot answer" response, he must be allowed to tell lies about whether he doesn't know/cannot answer.

If we allow the Knave the "I don't know/cannot answer" response, he always has two possible lies to any question. If the true answer is yes, he can say no or "I don't know/cannot answer." If the true answer is no, he can say yes or "I don't know/cannot answer." If the true answer is "I don't know/cannot answer." he can say yes or no. Does he chose his lies randomly or does he choose whichever lie makes the puzzle harder to solve?

One cannot help but wonder if the multiple posters who fail to read and understand the puzzle as stated are posting from rec.org.mensa...

No. The knave will also answer yes. The true answer in the case of the knave is no (If you asked him

1000 times if 2+2=4, he would not always answer yes). He would then lie by telling you yes.

The randomizer might answer either way, 50-50 chance, so it would be incorrect to say that only the knight will answer yes (implying that the randomizer will say no).

No. The randomizer might answer either way, 50-50 chance.

At least you read and understood the puzzle as stated...

That violates the "They will answer any yes/no question" clause.

One cannot help but wonder if the multiple posters who fail to read and understand the puzzle as stated are posting from rec.org.mensa...

"who will answer any yes/no question" clearly implies that no non yes/no questions are allowed. If you asked "If I was to ask this person who is the knave, who would he identify?" the ihabitants of the island would just look at you and wait for a yes/no question.

If they are allowed to answer any question, why not just ask "if you are the knight, please correctly identify knight, knave and randomizer. If you are the knave, please incorrectly identify knight, knave and randomizer."

Knight gives correct identities. Knave gives correct identities. Randomizer says "yes" or "no." If you get the randomizer, just ask someone else.

But you can't ask that questin and be sure of an answer. Nowhwere does the puzzle as stated define what the response to a non yes/no question to te knight or knave is, and the answer from randomizer is defined but give no information.

Not necessarily. The way the puzzle is worded the randomiser can, quite legitimately, answer either "yes" or "no" to this. It says he answers "yes" or "no" at random, not that he works out the answer to your question and then randomly inverts it.

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Neat! Cute! But then, only one question is enough:

1) Mr Knight... of the other two, is the [distinctive factor] a knave?

Distinctive factor depends on the actual setting, but here are some examples: "taller", "younger", "nearer to you", "next to you in clockwise direction", etc.

(If there is no distinctive factor for all pairs, then at least two of the three persons are the same person.)

- Risto -

Yes, it was quite late when I read through the thread and came up with my idea. But then as I "slept on it" I saw the fallacy of my assumption. If you could actually ask the randomizer 1000 times, the statistical results would be significant, but with only one question, it doesn't work.

Now suppose you asked one person if the other two would have a high probability (>60%) of answering yes to a known true statement.

If you asked the knave, the answer would be no, because the randomizer would give 50% yes and the knight would give 100% yes, so the probability would be 75%, and the knave would answer falsely.

If you asked the knight, the answer would be no, because the randomizer gives 50% no and the knave gives 100% no, so the probability is 75% no, or

25% yes, and the knight will answer truthfully no.

The randomizer would seem to give no worthwhile information.

But an answer of yes would absolutely identify the randomizer, in which case one more question of one of the others might identify all three. Once the randomizer is known, you can ask one of the others if that one is the randomizer, and you will know his identity from his answer. So there is a chance that two questions could suffice, but it would be only 1 in 6.

Now my brane hurts! Owww!

Paul

It wasn't raining It was raining, but their destination had radiators outside, which dried them off very quickly It was a really big umbrella or they were under a canopy the whole time The ladies' names were Gertrude Quitedry and Griselda Quitedry. They walked over a block to the BART station and then took BART to their destination, the San Francisco airport. By the time they got there they were dry "Dry" refers to the fact that they were abstaining from alcohol

Alan

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Defendit numerus

Hear ye, Hear ye. Naughty Knaves and Know Nots. Listen to a Knight who won't lead ye a stray. Let's recap how we got off Dichotomy Island finding the correct passage.

The sign said "Knaves Tell No Truth and Knights Tell No Lie, Pick the Wrong Passage a Ye Will Die"

and we're allowed only one question.

If we have a Question with a yes or no answer and we call the answer Q, then

Knight(Q)=Q and Knave(Q) =!Q

and Knight(Knave(Q))=Knave(Knight(Q))=!Q

Call our two unknown guards A and B, We asked A "If we asked B would he tell us to go through your passage." If he said yes we went through the opposite passage. If no we went through the one indicated in the question.

Now on to Trichotomy Island. The Sign Says

"If Your Life You Are to Save, Hurt No KnowNot, Befriend The Knight, And Kill The Knave"

and you may ask two of the three guards, one question each. Call our Guards A,B and C.

Question 1: We ask A, "If with our next question, we asked B if A would say C is the Know Not would he say, 'yes'" That is a yes or no question, but KnowNot(Q) =? so If B is the KnowNot and A is a Knight, The Knight can not give an answer which is always True. And if A is the Knave the Knave cannot give answer which is sure to be a Lie. If A is the Know Not than B is not the Know Not. So if you get an answer from A, you know B is not the Know Not. If you don't get an answer from A, B is the Know Not.

Question 2: a.We Know B is the KnowNot What we want here is Knight(Knave(Knight(Q)))=!Knave(Knight(Knave (Q))) Ask C if A would say C would say B is the KnowNot. Yes: C is the Knight, A is the Knave. No: A is the Knave, C is the Knight

b. We Know B is not the Know Not The First Question Asked is of the form A(B(A(Q))) which is Knight (Knave(Knight(Q))) =!Knave(Knight(Knave(Q))) so if C is the Know Not and the answer to the first question is yes than A is the Knave and B is the Knight, if the answer to the first question was no B is the Knave and A is the Knight.

So we want to ask B" if C would say B would say A is the KnowNot". No Answer C is the Know Not and A and B are as above. Yes: B is the Knave, C is the Knight. No: C is the Knave, B is the Knight.

So we kill the Knave and the Knight leads us to the passage to Quadotomy Island, which is populated by Knights, Knaves, Know Nots and Nymphs that never say No. There is no escape from the Nymphs that never say no.

Failing to understand would raise questions about their intelligence. But what would not reading it properly say? Are carelessness and intelligence incompatible attributes?

Sylvia.

The randomizer would answer yes or no.

The knight and knave must answer any yes/no question. It appears that they might or might not answer any other question, but if they answer, they would be constrained by their own rules of conduct, and so could not answer yes or not (since it's not a yes/no question), and a yes or no answer would neither be false nor true.

The net effect of this is that, as Willem has observed, a non yes/no question can only be used to determine whether the questionee is a randomizer or not. The randomizer answers yes or no, and the other two either say something else, or don't answer at all.

Sylvia.

wrote

What's worse, multiple posters or multiple postings of the same thing?

The original puzzle has already being dismissed as being impossible, one must wonder about posters who missed the original posters additional comment to adjust the puzzle to one that works.

Herc

Probably the intended solution, but it doesn't quite fit the problem as stated, which said that knights and knaves will answer any yes/no question. This seemed to preclude asking yes/no questions which a knight or knave cannot answer.

Sylvia.

We're not at this point asking P2 anything. The only question asked is of P1.

Sylvia.

They used to say you can't flunk an IQ test/

Guess what.2

This was an IQ test that many folks failed.