## Three gods - True, False and Random

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### Three gods - True, False and Random

This riddle is from a book called "Logic, Logic and Logic", where it is called "The Hardest Logical Puzzle Ever". It is a like a more difficult version of the three princesses puzzle, and knowing the solution to that one will certainly help.

There are three gods, True, False and Random. True will always answer truthfully, False will always lie, and Random will answer your questions randomly. The gods are standing before you, but you do not know which is which. Your task is to determine the identity of the gods, by asking them three questions. To make matters worse, the gods will answer your questions only with 'da' or 'ja', which mean true and false in their own language, but you don't know which is which.

Clarifications:
* Each question can be directed only at one god, but you may ask two (or three) of your questions to the same god, if you so wish
* The gods, being gods, are all-knowing and perfect logicians.

Torn Apart By Dingos

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Clarification: if I ask Random a nonsensical question, will I still get an answer?
Also: Are their actual names "True", "False", and "Random", as in if I asked True what the first letter of their names were, I'd get "T", "F", and "R" as an answer?
Also mk. 2: Are metaquestions permitted?

peri_renna

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The solution to this puzzle is so complicated that it scares me...
Spider

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peri_renna wrote:Clarification: if I ask Random a nonsensical question, will I still get an answer?
Also: Are their actual names "True", "False", and "Random", as in if I asked True what the first letter of their names were, I'd get "T", "F", and "R" as an answer?
Also mk. 2: Are metaquestions permitted?

Their only answers are "ja" and "da", and at least one always answers truthfully, and the actual question should be well formed, so I'd say there's an unwritten "by asking them three true/false questions". I'd also guess questions like "Will you answer with your word for 'false' or (make it known to me which of you is which and give me a billion dollars)?" are disallowed, because then you could ask all three that question and walk away with the answer and a billion dollars.
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GreedyAlgorithm

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GreedyAlgorithm wrote:
peri_renna wrote:Clarification: if I ask Random a nonsensical question, will I still get an answer?
Also: Are their actual names "True", "False", and "Random", as in if I asked True what the first letter of their names were, I'd get "T", "F", and "R" as an answer?
Also mk. 2: Are metaquestions permitted?

Their only answers are "ja" and "da", and at least one always answers truthfully, and the actual question should be well formed, so I'd say there's an unwritten "by asking them three true/false questions". I'd also guess questions like "Will you answer with your word for 'false' or (make it known to me which of you is which and give me a billion dollars)?" are disallowed, because then you could ask all three that question and walk away with the answer and a billion dollars.

I'm sorry, let me rephrase.

1. Suppose I don't know who is who. I ask the first one, "Would the one of your companions who isn't Random answer in the affirmative if I asked if that one [points] is Random?" It's an awful question, but it makes sense if I ask someone who isn't Random - and if I'm asking Random, then it's ambiguous. Would Random still randomly say "ja" or "da"? If so, would it be legal?

3. Can I ask otherwise-valid metaquestions (i.e. questions about questions), like "If I asked if you were True, would you answer in the affirmative?"

peri_renna

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peri_renna wrote:1. Suppose I don't know who is who. I ask the first one, "Would the one of your companions who isn't Random answer in the affirmative if I asked if that one [points] is Random?" It's an awful question, but it makes sense if I ask someone who isn't Random - and if I'm asking Random, then it's ambiguous. Would Random still randomly say "ja" or "da"? If so, would it be legal?

If you ask a question that doesn't have an answer, the gods will get angry and send a lightning bolt through you. But I suppose this could be allowed. You can think of Random as randomly blurting out an answer without even listening to your question.

Correct, but I don't see how this could be important. The gods all know their own and the others' identities. So you can refer to the god "True" and all the gods will know who you're talking about. They're all-knowing, remember?

3. Can I ask otherwise-valid metaquestions (i.e. questions about questions), like "If I asked if you were True, would you answer in the affirmative?"

I see no reason why not. It's not needed though. The solution I know about only uses pure logic questions.

Torn Apart By Dingos

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Thanks!

Torn Apart By Dingos wrote:
3. Can I ask otherwise-valid metaquestions (i.e. questions about questions), like "If I asked if you were True, would you answer in the affirmative?"

I see no reason why not. It's not needed though. The solution I know about only uses pure logic questions.

Crud. My solution's all metaquestions. I'll have to think on this.

peri_renna

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This one is making my brain hurt. Fooey.

koreanforrabbit

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Got it. I had to resort to aaronspook's method of proving the problem unsolvable, and then finding the flaw in the proof - but I've got a path to solution that uses no metaquestions.

It's about four times wordier than my metaquestions proof, just for the record. Horrible.

peri_renna

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Just to make sure I'm doing this right, do True, False and Random all speak the same language? As in if da means yes for True does it also mean yes for False and Random?
djooleek

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djooleek wrote:Just to make sure I'm doing this right, do True, False and Random all speak the same language? As in if da means yes for True does it also mean yes for False and Random?

I can't decide how best to respond sarcastically, and you're new and you've properly introduced yourself, so I'll just say...

Yes. Yes it does.

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I haven't actually solved this, but I'll still post a very basic hint for people who are stuck in the very early stages.

hint wrote:Lining up the gods in a row, we see there are six ways the gods could be arranged (True first, False second, Random third, etc.). Also, da could mean yes and ja could mean no, or vice versa. So all in all, there are twelve possibilities. Your three questions can only divide these twelve possibilities into eight groups. As such, by the Pigeonhole Principle, one such group must contain two possibilities. If you receive this set of three answers to your questions, then you won't be able to tell which of these two possibilities you're in, but you'll still have to be able to tell which god is which. As such, you must not be able to tell which of da and ja means yes. Specifically, this means that your solution to the riddle has to have a case where you figure out the identities of the gods but do not know which of da and ja means yes.
Cauchy

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How does random behave on metaquestions? Does it randomly decide to lie/not lie or does it just give a completely random answer?
Don't pay attention to this signature, it's contradictory.

Strilanc

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Solution wrote:Capitalizing on the last puzzle's solution, I offer this one.

I first introduce the "randomfinder" question, which goes as follows: I ask the person in the middle:

"Is the 'exclusive or' of the truth-values of the following two statements true? 'Ja' means yes. The person to your right is more likely to answer truthfully than the person on your left."

If the gods insist on playing politics with "your" right versus "my" right, then I'll either use something like North and South or just stand behind them, facing in the same way as them. But to clarify, the word "your" refers to the god's left and right.

Now, suppose the middle god is not R. Then, "ja" translates to "the random dude is on my left," whereas "da" translates to "the random dude is on my right." That's why I call this question the randomfinder question: It finds R. (If you don't believe me, say so and I'll go through the truth tables for you. But I'll assume for now that you can confirm this on your own.)

Now, of course, the middle person could be R himself, so we don't yet know who R is. One more question will give me that.

In the meantime, let's select someone on this grounds: If the middle god said "ja", select the god to his right. If he said "da," select the god to his left. We already know that if the middle god was not R, then this god you've selected is also not R. But we also know that if the middle god was R, then this guy is also not R -- because R is the middle god.

So, the god you've selected is either T or F, but definitely not R.

To save me from phrasing another question, we'll assume that you can move the gods' order as you wish. Move the god that you just selected to the center.

So ends question 1.

Since the god in the center is not R, ask him the randomfinder question.

Congratulations, you just found R.

So, that was question 2.

Question 3: ask the god in the middle, "does 'ja' mean 'yes'?"

Suppose he's a liar -- then he'll be forced to answer 'da,' so that he answers falsely regardless of whether 'ja' or 'da' means yes.

Suppose he's a truth-teller -- then he'll be forced to answer 'ja', so that he answers correctly regardless of whether 'ja' or 'da' means yes.

Congrats: You have just discovered which one R is, and which one the middle is. The last falls into place by the pigeonhole principle, obviously.

Looking back at Cauchy's hint, he makes a great point: At least in some of these cases, you do not actually get to learn whether "ja" or "da" maps to "yes" or "no."

Drostie

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Here are three other versions of the riddle in the original post:

THE OPERATORS OF BOOLANIA

There are three omniscient gods sitting in a chamber: AND, OR, and XOR. They all answer the truth, but they all apply their namesake operations in one of the following ways:

â€¢BACKWARD: The operator is applied to ALL of the questions that have been asked THUS FAR.
â€¢UNIVERSAL: The operator is applied to ALL of the questions (you have to pick your questions beforehand).
â€¢SELFLESS: The operator is applied to ALL of the questions NOT asked to them (you pick your questions beforehand).
For each of the 3 cases, determine with proof the minimum number of questions that will allow you to identify which god is which.

--------------------------------------------------------------------------------

Note 1: (Standard; rules that are generally assumed unless otherwise noted.) The gods only answer yes/no questions. Each god answers in the single word of their language as appropriate to the question; i.e. each god always gives one of only two possible responses, one affirmative and one negative (e.g. they would always answer "Yes" rather than "That would be true"). Each question asked must be addressed to a single specific god; asking one question to all the gods would constitute three questions. Asking a single god multiple questions is permissible. The question you choose to ask and the god you choose to address may be dynamically chosen based on the answers to previous questions.

Note 2: (Specific) Because of possible time conflicts, you must determine your questions ahead of time, rather than based on previous answers. However, you are still allowed to choose who you ask each of your three questions to dynamically. Scoping is also dynamic; e.g. the pronoun "you" in a question will always refer to the person to whom you are currently asking a question, not a predetermined person. No time related questions (e.g., "if the answer to my second question was 'no', then X otherwise Y") are permissible, as this could lead to paradoxes within the space-time continuum.

THE GODS OF GIBBERLAND

There are three omniscient gods sitting in a chamber: GibberKnight, GibberKnave, and GibberKnexus, the gods of the knights, knaves, and knexuses of Gibberland. Knights always answer the truth, knaves always lie, and knexuses always answer the XOR of what the knight and knave would answer.

Unfortunately, the language spoken in Gibberland is so unintelligible that not only do you not know which words correspond to "yes" and "no", but you don't even know what the two words that represent them are! All you know is that there is only one word for each.

With only three questions, determine which god is which.

--------------------------------------------------------------------------------

Note 1: What follows are standard rules that are generally assumed unless otherwise noted. The gods only answer yes/no questions. Each god answers in the single word of their language as appropriate to the question; i.e. each god always gives one of only two possible responses, one affirmative and one negative (e.g. they would always answer "Yes" rather than "That would be true"). Each question asked must be addressed to a single specific god; asking one question to all the gods would constitute three questions. Asking a single god multiple questions is permissible. The question you choose to ask and the god you choose to address may be dynamically chosen based on the answers to previous questions. No self-referential questions (e.g. "is this question true iff ...").

Note 2: Because of possible loop conflicts, you may not ask any questions regarding how a knexus would answer.

PAST, PRESENT, FUTURE

There are three omniscient gods sitting in a chamber: Past, Present and Future. They are all truthful, but with the following caveat: Present answers the question currently being asked, Past answers the last question asked in their chamber, and Future answers the next question which will be asked in their chamber. Despite their manipulation of which question to answer, each still answers immediately as if answering the question currently being asked.

Furthermore, the gods answer in a language in which "yes" and "no" are replaced by "da" and "ya", but you do not know which is which. You only know that their answers are consistent amongst themselves.

With three questions, determine which god is which.

--------------------------------------------------------------------------------

Patashu

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### Re: Three gods - True, False and Random

What a terrible curse, to be all-knowing but compelled to spit out random answers ... I bet the Ancient Greeks thought of this one.

MMoto

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Patashu wrote:THE OPERATORS OF BOOLANIA

There are three omniscient gods sitting in a chamber: AND, OR, and XOR. They all answer the truth, but they all apply their namesake operations in one of the following ways:

â€¢UNIVERSAL: The operator is applied to ALL of the questions (you have to pick your questions beforehand).

--------------------------------------------------------------------------------

Note 2: (Specific) Because of possible time conflicts, you must determine your questions ahead of time, rather than based on previous answers. However, you are still allowed to choose who you ask each of your three questions to dynamically. Scoping is also dynamic; e.g. the pronoun "you" in a question will always refer to the person to whom you are currently asking a question, not a predetermined person. No time related questions (e.g., "if the answer to my second question was 'no', then X otherwise Y") are permissible, as this could lead to paradoxes within the space-time continuum.

It seems to me like this dynamic scoping under universal application leads to ruptures in the spacetime continuum just as easily. As a simple example, suppose there are only two gods, AND and OR, and I ask only two identical questions: "Are you the AND god?" and "Are you the AND god?". If the answer to my first question is yes, I ask the same god the second question. If it's no, I ask the other god the second question. Now, what happens if I start with the OR god?

(I realize that I don't know which god is the OR god, and I don't know which response is yes. So I can't actually do this strategy and guarantee that it leads to a paradox: I'm just showing that there is some strategy that does.)

Cosmologicon

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As with all logic puzzles, if doing something leads to a contradiction or contradictory behavior, it's implied that you aren't allowed to do it.

Patashu

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Cosmologicon wrote:
Patashu wrote:THE OPERATORS OF BOOLANIA

There are three omniscient gods sitting in a chamber: AND, OR, and XOR. They all answer the truth, but they all apply their namesake operations in one of the following ways:

â€¢UNIVERSAL: The operator is applied to ALL of the questions (you have to pick your questions beforehand).

--------------------------------------------------------------------------------

Note 2: (Specific) Because of possible time conflicts, you must determine your questions ahead of time, rather than based on previous answers. However, you are still allowed to choose who you ask each of your three questions to dynamically. Scoping is also dynamic; e.g. the pronoun "you" in a question will always refer to the person to whom you are currently asking a question, not a predetermined person. No time related questions (e.g., "if the answer to my second question was 'no', then X otherwise Y") are permissible, as this could lead to paradoxes within the space-time continuum.

It seems to me like this dynamic scoping under universal application leads to ruptures in the spacetime continuum just as easily. As a simple example, suppose there are only two gods, AND and OR, and I ask only two identical questions: "Are you the AND god?" and "Are you the AND god?". If the answer to my first question is yes, I ask the same god the second question. If it's no, I ask the other god the second question. Now, what happens if I start with the OR god?

(I realize that I don't know which god is the OR god, and I don't know which response is yes. So I can't actually do this strategy and guarantee that it leads to a paradox: I'm just showing that there is some strategy that does.)

Hmm... the way I first read the rules, that doesn't count as a paradox. Since scoping is dynamic, the OR god will apply his operation to both the questions as if they were both addressed to him, leading him to simply answer "no".
Token

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I am wondering, do you always ask the same three questions?

i.e.

I ask question 1 of some god and get reply Da (say). I don't know if Da means "yes" or "no", so there is no point modifying my strategy based on this answer - I may as well have decided on what question 2 is (and who its to) before even asking question 1.

I ask question 2 and get a reply, (either Da, same as before, or Ja, different to before). Does my choice of what question 3 to ask (and who to ask) depend on whether the answer to question 2 is the same or different to question 1?

Just wondering.

(very stuck)

mable
mable

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Have a go at three princesses first mable, its a good intro to this kind of problem.

http://forums.xkcd.com/viewtopic.php?t=87

and I'm pretty sure you're allowed to change the questions and who you ask based on earlier answers.

jestingrabbit

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this is not a solution but rather me asking for help on clarifying the solution. it does reveal how the problem is solved however.

I had this down to knowing that the first two questions must tell me who the random god is using logic similar to the princess question and that the third question must be does da mean yes, before looking at Drosties solution to figure out the exact wording of that first question. however english being my worst subject I still have no clue what that question means or why it works. Im not sure what "is the "exclusive or" of the truth-values of the following two statements true" means". could somebody clarify that for me.

Ansain

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Im not sure what "is the "exclusive or" of the truth-values of the following two statements true" means". could somebody clarify that for me.
Exclusive or:
Code: Select all
TF
T|FT
F|TF
That is, it returns true when one of the two statements compared is true, and the other is false, and false when both are true or both are false. This is different from a regular or, where both being true will return true.
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Vaniver

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thanks, that makes a whole lot more sense to me now.

Ansain

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I ask one of them Is my name karl?(my name IS karl) i suppose he would say ja. I would ask another question where i know that is true(ex. im a boy). Then i ask the one who keeps saying da a question where i know it is not true, if he says ja, then i would suppose he is trying to tell me lies. I would then suppose that ja is the answer for yes. Then i ask the last god, I ask him questions where I know every answer is true and if he keeps answering da or ja, i would suppose he is random.

Example

Me: Is my name karl?
god 1: ja
me:am i a boy?
god 1: ja
me am i human?
god 1: ja

me: Is my name karl?
god 2: da
me: am i a boy?
god 2: da
me: am i human?
god 2: da

me: is my name karl?
god 3: da
me: am i a boy?
god 3: ja
me:am i human?
god 3: da

i would then suppose god 1 as true, god 2 is false, god 3 is random

you could interchange the conversations depending on what happens when you ask on of them.
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ze_heineken

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ze_heineken wrote:i would then suppose god 1 as true, god 2 is false, god 3 is random

you could interchange the conversations depending on what happens when you ask on of them.

There is always the risk of getting all jas or all das on the random one though.
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ze_heineken wrote:stuff

The whole point of the puzzle is to be able to work out the identities of the gods using only three yes or no questions, each question asked to only one god. Anyone could do it if you're allowed an unlimited number.

jestingrabbit

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I re-read Drostie's solution, and understood it that time, really cool solution.

Now I see from Drostie's first question that what I said was not true -

I was imaginging that there was no point choosing who to direct the second question to, based on the answer to the first question.

But as shown in the solution, that is exactly what you do (have to?) do.

Nice one Drostie.
mable

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torn apart by dingos wrote:
but you may ask two (or three) of your questions to the same god
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ze_heineken

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There are only three questions in total.

Torn Apart By Dingos

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I'm thinking you have to ask a god about the other two in some way, eg
"Is that god the god that tells the truth?"

Intuitively, i dont think its possible to get the answer in 3 questions by not invoking the other two in some way when talking to one of them. That make sense?

dbsmith

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### Re:

ze_heineken wrote:I ask one of them Is my name karl?(my name IS karl) i suppose he would say ja. I would ask another question where i know that is true(ex. im a boy). Then i ask the one who keeps saying da a question where i know it is not true, if he says ja, then i would suppose he is trying to tell me lies. I would then suppose that ja is the answer for yes. Then i ask the last god, I ask him questions where I know every answer is true and if he keeps answering da or ja, i would suppose he is random.

Example

Me: Is my name karl?
god 1: ja
me:am i a boy?
god 1: ja
me am i human?
god 1: ja

me: Is my name karl?
god 2: da
me: am i a boy?
god 2: da
me: am i human?
god 2: da

me: is my name karl?
god 3: da
me: am i a boy?
god 3: ja
me:am i human?
god 3: da

i would then suppose god 1 as true, god 2 is false, god 3 is random

you could interchange the conversations depending on what happens when you ask on of them.

This would be an excellent way of determining the answer if you were able to ask nine questions.

Shark3226

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### Re: Re:

Shark3226 wrote:
ze_heineken wrote:I ask one of them Is my name karl?(my name IS karl) i suppose he would say ja. I would ask another question where i know that is true(ex. im a boy). Then i ask the one who keeps saying da a question where i know it is not true, if he says ja, then i would suppose he is trying to tell me lies. I would then suppose that ja is the answer for yes. Then i ask the last god, I ask him questions where I know every answer is true and if he keeps answering da or ja, i would suppose he is random.

Example

Me: Is my name karl?
god 1: ja
me:am i a boy?
god 1: ja
me am i human?
god 1: ja

me: Is my name karl?
god 2: da
me: am i a boy?
god 2: da
me: am i human?
god 2: da

me: is my name karl?
god 3: da
me: am i a boy?
god 3: ja
me:am i human?
god 3: da

i would then suppose god 1 as true, god 2 is false, god 3 is random

you could interchange the conversations depending on what happens when you ask on of them.

This would be an excellent way of determining the answer if you were able to ask nine questions.

Not really - the random god could just as easily choose to answer all da's or all ja's
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### Re: Re:

Keand64 wrote:
Shark3226 wrote:This would be an excellent way of determining the answer if you were able to ask nine questions.

Not really - the random god could just as easily choose to answer all da's or all ja's

What 3 questions would you ask when determining whether to commit a 3 year necro?
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mike-l

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### Re: Three gods - True, False and Random

I haven't read any replies due to lack of patience and sleep., so if someone has already solved this, sorry for the echo.

Spoiler:
I'll call each god of unknown identity A, B, and C, and use the names "Angel", "Demon" and "Chaos" instead of OP's names, to avoid confusion with other words.

When you realize that saying "Is it true that both or neither statements 'ja means yes' and '[statement]' are true?" means that the god will respond "ja" if they would tell you that [statement] is true and "da" if the would tell you [statement] is false, you don't need to know if ja and da mean yes or no. It is impossible to know that and the identity of all the gods in this puzzle. You have only 8 combinations of questions and 12 possible outcomes if you have to find the meaning of ja and da.

Also, if we prefix our question with "What would you say if I were to ask you...", then Demon will answer the same way as Angel, because he must lie about the his hypothetical lie (like a double negative.)

The first question is will be directed to A.

Question 1: What would you say if I were to ask you if it is true that both or neither statements "ja means yes" and "B is not Chaos" are true?

This is to make sure that our next question isn't directed to Chaos. If A is Chaos, it doesn't matter what he answers, because we aren't talking to A again. Otherwise, both Angel and Demon will answer "ja" for B is Chaos, in which case the last two questions will be directed to C, otherwise, the last two questions will be directed to B.

Question 2: What would you say if I were to ask you if it is true that both or neither statements "ja means yes" and "You are Demon" are true?

Question 3: What would you say if I were to ask you if it is true that both or neither statements "ja means yes" and "A is Chaos" are true?

Now, with those questions, the following table gives the identity of each god for each combination of questions:

Code: Select all
(except for the header, A=Angel, D=Demon, C=Chaos)

Q1 Q2 Q3 A B C
ja ja ja C A D
ja ja da A C D
ja da ja C D A
ja da da D C A
da ja ja C D A
da ja da A D C
da da ja C A D
da da da D A C
Blue, blue, blue

undecim

Posts: 286
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### Re: Three gods - True, False and Random

Spoiler:
If the answer is "ja", then A is not random, and if the answer is "da", B is not random. For the sake of argument, assume that C says "ja". We now know that B is the random god, and A and C are true and false, but not necessarily respectively. Conclude by asking A: "Does 'da' mean 'true'?"
Remixen

Posts: 1
Joined: Tue Aug 17, 2010 5:04 am UTC

### Re: Three gods - True, False and Random

POST DELETED DUE TO UNKINDNESS - NEVERMIND
Last edited by t1mm01994 on Tue Aug 17, 2010 3:26 pm UTC, edited 1 time in total.

t1mm01994

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Location: San Francisco.. Wait up, I'll tell you some tales!

### Re: Three gods - True, False and Random

On these forums we're fine with necroing. Back seat moderating... not so much.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

jestingrabbit

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Location: Sydney

### Re: Three gods - True, False and Random

Coming rather late to this party...here's a hopefully shorter and clearer answer, seems equivalent to Drostie's, but I only skimmed it.

Spoiler:
To rephrase what Cauchy said, information theory shows that we can't find out what "da" and "ja" mean. You can ask three T/F questions, which can yield a maximum of 3 bits = 8 states. There are 6 permutations of the gods. To discover additionally what "da" and "ja" mean would require 12 states worth of information. Thus, we must ask all our questions in a way that ignores the meaning of these words, such as "is the answer 'da'".

Q1. Ask God #1 "If I asked God #2 and God #3 if 2+2=4, God #2 would be more likely to answer 'da'."
"Da" means God #1 claims that God #2 is more truthful than God #3. "Ja" means the opposite.
Case 1: God #1 is True; will say the more truthful is Random, less is False.
Case 2: God #1 is False; will say the more truthful is Random, less is True.
Case 3; God #1 is Random; God #2 and God #3 are True and False but we don't know which is which.

We now know that the one called less truthful is either True or False. Let X = that god's number and Y = the number of the god called more truthful.

"Da" means X is True; "ja" means X is False.

Q3: Ask God X "If I asked God #1 and God #Y if 2+2=4, God #1 would be more likely to answer 'da'."
If X is True, the true answer is what he says. If X is False, the true answer is the opposite.
Case 1: the true answer is "da". This means God #1 is the more truthful of the remaining gods.
Case 2: the true answer is "ja". This means God #1 is the less truthful of the remaining gods.

Tualha

Posts: 61
Joined: Wed Dec 12, 2007 12:18 pm UTC

### Re: Three gods - True, False and Random

I suppose, if these God's are all knowing, then they would know my name.

so the easiest way to break down the three would be to ask them all on separate occasions "Is my name ______" to which one would, without question answer yes, that is your name and one would definitely answer no and the third would either answer yes or no.
If the random God answers yes, it would be a matter of asking a series of obvious, entirely tangible questions until chance played it's part and the random God lied. Seemed pretty simple, really....
Slaggagary

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