The Gods of Logic

[ThunderCloud]

This puzzle is a variation on the "Hardest Logic Puzzle Ever", as can be seen on Wikipedia. I've found it to be a fun problem to play with.

There are three gods, each of whom speaks through his respective totem. One god always tells the truth, one always lies, and one answers entirely at random. The three totems are unlabeled, so you do not know which god is which. The gods respond only to yes-no questions, and may only be addressed individually via the querant's choice of totem. Furthermore, each god answers in his own personal language, and you know nothing in advance about any of the three gods' languages, save that each includes distinct words for "yes" and "no". Your task is to correctly ascribe each totem to its respective god with only three questions. What are your questions, and how are they directed?

Note: Because the "god of Truth" must always tell the truth, and the "god of Lies" must always lie, neither god is able to respond to a question which lacks a definite answer. The "god of Randomness", however, will respond to any question -- his response is unrelated to the content of the question, but is instead prompted by the fact that he was asked one.

[Yoruichi-san]

Um...I'm getting deja vu...is it something like

[ThunderCloud]

Um...I'm getting deja vu...is it something like

Something like, but not exactly like. It is a variation. Each god speaks his own individual language (so there are three unknown languages), and you do not know the words each will use in advance.

[bonanova]

Y-san, I see two complications over the standard "hardest logic puzzle."

1. The gods do not respond da or ja: each god has his own language.
   
2. The third god does not randomly lie or tell the truth, he randomly says yes or no.
   He may thus answer any question.

ThunderCloud, you don't specifically limit our questions to be of the Yes/No variety, but it seems implied?

[Yoruichi-san]

I see...

Actually the puzzle I linked had the same setup for 2. @bon-chan.

A couple questions to TC:

1. Do the other two gods know what Random is going to answer before hand?

2. Are you allowed to ask questions that potentially do not have a definite answer? Can you take the totem's lack of an answer as an information source?

3. Are you allowed to ask questions that potentially create a paradox?

If you ask "Do you always lie?" to the lying god, does he not answer or his head explode ;P?

You could potentially use a strategy of using non-answers to ferret out the random god with the first two questions and the lying god with the last.

[phaze]

4. If I were to ask the lying god about a question the truth telling god would have not been able to answer is it possible for him to lie by answering

Asking the lying god

If the god to you left (who just happen to be the truth telling God) was asked if his answer would mean no what would he reply.

Now the truth telling god cannot reply to the question "will your answer mean no". Can the lying god whom you are addressing then say his equivalent of "Yes" or "No" as he chooses (and thus lying) or would he also be constrained not to answer?

[Thalia]

Hope I didn't mix up the questions after rearranging them a billion times. . .

If non yes/no questions are allowed. . .

1. To #1: What is 2+2?

-Answer- #1 is random answerer- See question 4.
-No answer- See question 2.
2. To #1: If I ask #2 if 2+2=4, what will he say?
-No answer-#2 is random answerer- See question 3.
-Answer- The word given in answer means no.- See question 3.
3. To #1: Does 2+2=4?
-Answers same as before- #1 is liar- if no answer to question 2, 2 is random answerer, #3 is truth teller. If answer to #2, switch #2 and #3.
-Answers with different word- #1 is truth teller- if no answer to question 2, #2 is random answerer, #3 is liar. If answer to #2, switch #2 and #3.
4. To #2: Are you the truth teller?- Given answer means yes. -See question 5.
5. To #2: Does 2+2=4
-Answers same as before- #1 is random answerer, #2 is truth teller, #3 is liar.
-Answers with different word- #1 is random answerer, #2 is liar, #3 is truth teller.

Edited February 18, 2013 by Thalia

[ThunderCloud]

I see...

Actually the puzzle I linked had the same setup for 2. @bon-chan.

A couple questions to TC:

1. Do the other two gods know what Random is going to answer before hand?

2. Are you allowed to ask questions that potentially do not have a definite answer? Can you take the totem's lack of an answer as an information source?

3. Are you allowed to ask questions that potentially create a paradox?

If you ask "Do you always lie?" to the lying god, does he not answer or his head explode ;P?

You could potentially use a strategy of using non-answers to ferret out the random god with the first two questions and the lying god with the last.

In order:

(1) No.

(2) Yes.

(3) I am not sure what you mean...

If you ask the "god of Lies" the question "Do you always lie?", the question has a definite answer (i..e it is Yes), therefore the god will answer with his word for "No."

Such a strategy is what I am seeking.

[ThunderCloud]

4. If I were to ask the lying god about a question the truth telling god would have not been able to answer is it possible for him to lie by answering

Asking the lying god

If the god to you left (who just happen to be the truth telling God) was asked if his answer would mean no what would he reply.

Now the truth telling god cannot reply to the question "will your answer mean no". Can the lying god whom you are addressing then say his equivalent of "Yes" or "No" as he chooses (and thus lying) or would he also be constrained not to answer?

In this case he would not answer. Neither the "god of Lies" nor the "god of Truth" can respond to a question to which there is no definite answer.

[ThunderCloud]

Hope I didn't mix up the questions after rearranging them a billion times. . .

If non yes/no questions are allowed. . .

1. To #1: What is 2+2?

-Answer- #1 is random answerer- See question 4.

-No answer- See question 2.

2. To #1: If I ask #2 if 2+2=4, what will he say?

-No answer-#2 is random answerer- See question 3.

-Answer- The word given in answer means no.- See question 3.

3. To #1: Does 2+2=4?

-Answers same as before- #1 is liar- if no answer to question 2, 2 is random answerer, #3 is truth teller. If answer to #2, switch #2 and #3.

-Answers with different word- #1 is truth teller- if no answer to question 2, #2 is random answerer, #3 is liar. If answer to #2, switch #2 and #3.

4. To #2: Are you the truth teller?- Given answer means yes. -See question 5.

5. To #2: Does 2+2=4

-Answers same as before- #1 is random answerer, #2 is truth teller, #3 is liar.

-Answers with different word- #1 is random answerer, #2 is liar, #3 is truth teller.

There is a path that does not quite complete: If in (1) there is no answer, and in (2) there is no answer, then in (3) there can be no comparison to "the same answer as before."

[Yoruichi-san]

It does say in the OP that "the gods respond only to yes-no questions"...(paragraph 2, 4th sentence)

I meant to ask, what happens if you ask the lying god, "Is your answer to this question a lie?"

True would answer "no" definitely, but it's a paradox for Lie.

Ask two of them, "will the random god answer whatever means 'yes' in his language to the next question?", only Random can answer, so this tells you who Random is.

Use last question as above if acceptable.

[ThunderCloud]

It does say in the OP that "the gods respond only to yes-no questions"...(paragraph 2, 4th sentence)

I meant to ask, what happens if you ask the lying god, "Is your answer to this question a lie?"

True would answer "no" definitely, but it's a paradox for Lie.

Ask two of them, "will the random god answer whatever means 'yes' in his language to the next question?", only Random can answer, so this tells you who Random is.

Use last question as above if acceptable.

This is a different solution than the one I had in mind, and I like it.

I had not considered exploiting that aspect of the lying god...

Having said that, I know at least one solution that does not rely on this technique. So, to make life difficult: suppose that the lying god, when thusly cornered, answers at random and considers either response to be a "lie". Can you still identify the gods?

[ThunderCloud]

ThunderCloud, you don't specifically limit our questions to be of the Yes/No variety, but it seems implied?

If I understand your question correctly: the gods of Truth and Lies will not respond to questions for which an appropriate answer is neither "Yes" nor "No." The "god of Randomness" will answer any question with a "Yes" or "No" response.

[Yoruichi-san]

Lol, well as an engineer, I do try to be as lazy *cough* efficient as possible...

Denoting the gods as A,B, and C for simplicity...

1st question: Ask A, "Would B answer yes (or 'whatever in his language means yes' if you need to be that specific) to the question 'Is C the random god?'"

If he can't answer, then B is the random god. In this case, there are a couple ways you could go. The simplest IMO is probably to ask A, like, "Would you answer yes if I asked you 'Is the sky blue'?" or something like that that tells you which of his answers means yes (or no), and then for the last question ask "Is the sky blue" to determine which of A and C is the liar and truthteller.

If A did answer the first question, then either A or C is the random god, and B is definitely not the random god. In this case, take note of what he just said, and move on to B. Ask B, "Would C answer yes (or 'whatever in his language means yes') to the question 'Did A just say [insert whatever A just said]'?" (or 'Is the sky blue' or any question that is true I suppose...)

If B can't answer question 2, then C is the random god, and whatever A answered previously means 'no' in A's language. So move back to A and ask him 'Is the sky blue' or some such to determine which of A and B is the liar and the truthteller.

If B did answer question 2, A is the random god and whatever B just answered means 'no'. So then ask him 'Is the sky blue' to determine which of B and C is the liar and the truthteller.

(Of course, if you want to be technical you can always replace 'Is the sky blue' with a more undisputable mathematical question, like "does 1+1=2?" or logical question, like "Does yes mean yes?", but I just like to use it in these problems )

Edited February 19, 2013 by Yoruichi-san

[Thalia]

"Random"=R=Random answerer

T=Truth teller

L=Liar

1. To #1: If I ask 2 if he is random, what will he say?

-Answers- 2 is not random- go to question 4
-No answer- 2 is random- go to question 2

2. To #1: If I ask 3 if 2+2=4, what will he say?
-Answer will be “no”- go to question 3

3. To #1: Does 2+2=4?
-Answers same as question 2- 1L, 2R, 3T
-Answers different from question 2- 1T, 2R, 3L

4. To #2: If I ask 3 if he is random, what will he say?
-Answers- 1 is random- go to question 5
-No answer- 3 is random- go to question 6

5. To #2: Does 2+2=4?
-Answers same as question 5- 1R, 2T, 3L
-Answers different from question 5- 1R, 2L, 3T

To #2: Does 2+2=4?
-Answers same as question 1- 1L, 2T, 3R
-Answers different from question 1- 1T, 2L, 3R

Edited February 19, 2013 by Thalia

[phaze]

1. To #1: If I ask 2 if he is random, what will he say?

-Answers- 2 is not random- go to question 4

You may want to take note of what the answer is as it does not cost you anything but allows you to ask questions such as "Does Blarg mean yes?"

[ThunderCloud]

"Random"=R=Random answerer

T=Truth teller

L=Liar

1. To #1: If I ask 2 if he is random, what will he say?

-Answers- 2 is not random- go to question 4

-No answer- 2 is random- go to question 2

2. To #1: If I ask 3 if 2+2=4, what will he say?

-Answer will be “no”- go to question 3

3. To #1: Does 2+2=4?

-Answers same as question 2- 1L, 2R, 3T

-Answers different from question 2- 1T, 2R, 3L

4. To #2: If I ask 3 if he is random, what will he say?

-Answers- 1 is random- go to question 5

-No answer- 3 is random- go to question 6

5. To #2: Does 2+2=4?

-Answers same as question 5- 1R, 2T, 3L

-Answers different from question 5- 1R, 2L, 3T

To #2: Does 2+2=4?

-Answers same as question 1- 1L, 2T, 3R

-Answers different from question 1- 1T, 2L, 3R

The algorithm is self-referential in question 5 (probably a typo)...

[ThunderCloud]

Lol, well as an engineer, I do try to be as lazy *cough* efficient as possible...

Denoting the gods as A,B, and C for simplicity...

1st question: Ask A, "Would B answer yes (or 'whatever in his language means yes' if you need to be that specific) to the question 'Is C the random god?'"

If he can't answer, then B is the random god. In this case, there are a couple ways you could go. The simplest IMO is probably to ask A, like, "Would you answer yes if I asked you 'Is the sky blue'?" or something like that that tells you which of his answers means yes (or no), and then for the last question ask "Is the sky blue" to determine which of A and C is the liar and truthteller.

If A did answer the first question, then either A or C is the random god, and B is definitely not the random god. In this case, take note of what he just said, and move on to B. Ask B, "Would C answer yes (or 'whatever in his language means yes') to the question 'Did A just say [insert whatever A just said]'?" (or 'Is the sky blue' or any question that is true I suppose... )

If B can't answer question 2, then C is the random god, and whatever A answered previously means 'no' in A's language. So move back to A and ask him 'Is the sky blue' or some such to determine which of A and B is the liar and the truthteller.

If B did answer question 2, A is the random god and whatever B just answered means 'no'. So then ask him 'Is the sky blue' to determine which of B and C is the liar and the truthteller.

(Of course, if you want to be technical you can always replace 'Is the sky blue' with a more undisputable mathematical question, like "does 1+1=2?" or logical question, like "Does yes mean yes?", but I just like to use it in these problems )

Well done. My answer was very similar to this.

The only real difference was my third question in the case where B is not random. After determining which of (A, C) is random using question #2, I then ask the non-random god whose answer I have heard (either A or B) "If I were to ask you whether you always told the truth, would you say (insert word that this god has previously uttered)?" If he replies with the same word again, then he is the god of Truth; else he is the god of Lies.

Although it is possible to decipher at least one god's language during the course of the three questions (as you demonstrated), it is actually not necessary... which leaves me with the feeling that the problem can be made still harder, somehow....

[Yoruichi-san]

Yeah deciphering is not necessary, but it makes it mentally easier for me, at least, lol, to keep track of, and somehow I feel more accomplished for getting an extra piece of information ;P.

Check out the puzzle I linked...the setup there (where True and Lie can also answer any yes-no question...they are omniscient apparently about Random's answers) disallows you to know which word stands for which, since there were only two possible answers. In these puzzles 8 possible arrangements of True/Lie/Random, so 2^3=8 will resolve which arrangement with no information to spare. This puzzle there are three potential answers: a word, a word that is different than the previous word, and lack of an answer, so there is more possible information gained.

Nice puzzle tho, it was fun, thanks .

Edited February 19, 2013 by Yoruichi-san

[ThunderCloud]

Yeah deciphering is not necessary, but it makes it mentally easier for me, at least, lol, to keep track of, and somehow I feel more accomplished for getting an extra piece of information ;P.

Check out the puzzle I linked...the setup there (where True and Lie can also answer any yes-no question...they are omniscient apparently about Random's answers) disallows you to know which word stands for which, since there were only two possible answers. In these puzzles 8 possible arrangements of True/Lie/Random, so 2^3=8 will resolve which arrangement with no information to spare. This puzzle there are three potential answers: a word, a word that is different than the previous word, and lack of an answer, so there is more possible information gained.

Nice puzzle tho, it was fun, thanks .

I will take a look at it, thanks. Glad you enjoyed the problem. Btw, I only count 6 possible arrangements for True/Lie/Random (3!) ...?

[Yoruichi-san]

Lol...you're right...I got some stuff mixed up in my mind...too much on my mental plate...(which is why it's useful to know which word is 'no'! )...and the usefulness of the information is limited by the fact that Random's answers are not that informative...ahh head hurts...anyways, you get the point ;P.

[Thalia]

"Random"=R=Random answerer

T=Truth teller

L=Liar

1. To #1: If I ask 2 if he is random, what will he say?

-Answers- 2 is not random- go to question 4

-No answer- 2 is random- go to question 2

2. To #1: If I ask 3 if 2+2=4, what will he say?

-Answer will be “no”- go to question 3

3. To #1: Does 2+2=4?

-Answers same as question 2- 1L, 2R, 3T

-Answers different from question 2- 1T, 2R, 3L

4. To #2: If I ask 3 if he is random, what will he say?

-Answers- 1 is random- go to question 5

-No answer- 3 is random- go to question 6

5. To #2: Does 2+2=4?

-Answers same as question 5- 1R, 2T, 3L

-Answers different from question 5- 1R, 2L, 3T

To #2: Does 2+2=4?

-Answers same as question 1- 1L, 2T, 3R

-Answers different from question 1- 1T, 2L, 3R

The algorithm is self-referential in question 5 (probably a typo)...

Yes. That should say 4, not 5. . . I think. . . Not ready to remake the spreadsheet.

Edited February 20, 2013 by Thalia

[Thalia]

1. To #1: If I ask 2 if he is random, what will he say?

-Answers- 2 is not random- go to question 4

You may want to take note of what the answer is as it does not cost you anything but allows you to ask questions such as "Does Blarg mean yes?"

It would allow me to ask such questions. I'm not quite convinced that learning their language is necessary to determining the answer to the OP. Actually, as Y-San has demonstrated, I'm pretty sure it isn't.

[phaze]

@Thalia I think that "Did A just say [whatever A said]?" qualifies

[Thalia]

@Thalia I think that "Did A just say [whatever A said]?" qualifies

I'm not quite seeing the connection there. But I've also decided that I'll go crazy if I try to go through Y-san's explanation or this again.