Navigating the Path of Life: One Question to Find the Way

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In this discussion, participants explore a logic puzzle involving two gnomes, one who tells the truth and one who lies, guarding paths to heaven and hell. The challenge is to formulate a single question that reveals which path leads to heaven. Various proposed questions include asking one gnome what the other would say about the path he guards, which would effectively cancel out the lies. The conversation highlights the intricacies of logic and reasoning in determining the correct path based on the gnomes' responses. Ultimately, the focus remains on finding a valid question that can lead to the truth despite the presence of deception.
  • #31
Evo said:
No, for example, if the lier is on the path to heaven, he would say no, so the truthful one would be on the path to hell and also say no. You've got 2 gnomes both saying no, so you have no idea which path they are on. Same if they were on opposite paths, they'd both say yes.

There is one gnome standing in front of a door. The door leads to heaven or hell and the gnome always lies or always tells the truth. There is one question (and variations of it) that will determine all of the variables. (edit - not true. It only determines where the door leads. If the gnome is honest or a liar is undetermined, but irrelevent.)

So I ask him, "If I asked you where this door leads, would you tell me it leads to heaven?"

Ok, assume the door leads to heaven.
If someone asked the truthful gnome if the door went to heaven he would answer 'yes,' so his answer is 'yes.'
If someone asked the liar if his door leads to heaven he would answer 'no,' so his answer to me will be 'yes.'

In either case I can be sure that this is indeed the door to heaven. If I ask the same question and the answer is 'no' then I know the door leads to hell. The question isn't asking where the door leads. The question is asking what they would say if asked another question. In this logic argument two truthful statements combined is still a truthful statement, and two lies cancel each other out and become a truthful statement.
 
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  • #32
Huckleberry said:
There is one gnome standing in front of a door. The door leads to heaven or hell and the gnome always lies or always tells the truth. There is one question (and variations of it) that will determine all of the variables.

So I ask him, "If I asked you where this door leads, would you tell me it leads to heaven?"

Ok, assume the door leads to heaven.
If someone asked the truthful gnome if the door went to heaven he would answer 'yes,' so his answer is 'yes.'
If someone asked the liar if his door leads to heaven he would answer 'no,' so his answer to me will be 'yes.'

In either case I can be sure that this is indeed the door to heaven. If I ask the same question and the answer is 'no' then I know the door leads to hell. The question isn't asking where the door leads. The question is asking what they would say if asked another question. In this logic argument two truthful statements combined is still a truthful statement, and two lies cancel each other out and become a truthful statement.

there is a point there. so most likely, you will still go through thedoor right?:rolleyes:
 
  • #33
Rogerio said:
I would just say:
"If someone asked you whether your path leads to heaven, would you answer yes?"
And the answer would be true.

It is OK.

Evo said:
No, for example, if the lier is on the path to heaven, he would say no, so the truthful one would be on the path to hell and also say no.

Wrong.
According to your example, if someone had asked the liar gnome, the gnome would have answered "no" . So, that gnome would say "yes" to Rogerio.

BTW, this is a classical puzzle (google: liar truth teller).
Jimmysnyder and Rogerio are absolutely right.
 
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  • #34
Evo said:
...This isn't hard to follow guys.

I think so but... are you sure of that?
 
  • #35
kaisxuans said:
there is a point there. so most likely, you will still go through thedoor right?:rolleyes:

I didn't complete the comparison because I thought it was obvious.


Ok, assume the door leads to heaven.
If someone asked the truthful gnome if the door went to heaven he would answer 'yes,' so his answer is 'yes.'
If someone asked the liar if his door leads to heaven he would answer 'no,' so his answer to me will be 'yes.'

Ok, now assume the door leads to hell.
If someone asked the truthful gnome if the door went to heaven he would answer 'no,' so his answer is 'no.'
If someone asked the liar if his door leads to heaven he would answer 'yes,' so his answer to me will be 'no.'

The way the question is phrased it is irrelevant if the gnome is honest or if he is lying. Which door I would choose is also irrelevent, but at least I would be able to make an informed choice.
 
  • #36
I think the misunderstanding here is in the interpretation of the question.

"If I asked you where this door leads, would you tell me it leads to heaven?"

The question is not the same as "Does this door lead to heaven?" The original question is not asking directly where the door leads. It is asking what the gnome would say if asked another question. The lying gnome would lie if someone asked where his door led, and he would also lie to me about what he would say if someone asked him where his door led. So if his door led to heaven and someone asked him "Does this door lead to heaven?" then he would answer 'no.' If I ask him what his answer to that question would be then he would answer 'yes.' So in this scenario the answer to the original question, "If I asked you where this door leads, would you tell me it leads to heaven?" the answer is 'yes.'

In order to answer the question, the gnome must answer two questions. First he must determine what he would say if someone asked him where his door led. Then he must respond to the question asking him what he would say. The lying gnome would lie if asked where his door leads and would lie to me about his response. The lying gnome would be lying about a lie, and with the only options being 'yes' or 'no' his answer will be the truth.

Ofcourse, the honest gnome tells the truth no matter how many times he is asked a question. He would tell the truth if asked where his door leads and he would tell the truth if I asked him how he would answer that question. His answer will always be the truth because he doesn't contradict himself with lies.

So it is irrelevant if the gnomes are always honest or always lying.
 
  • #37
Evo said:
jimmysnyder said:
This in answer to the question "What would the other gnome say is the path he guarding?"
Well this is convoluted isn't it. It's not a yes/no question. But let's edit the question to fit the discussion.

Would the other gnome say he is guarding the path to heaven?

Only one of the gnomes is a liar. If the liar was on the path to heaven, he would say no. But then the other gnome is not a liar so he faithfully reports the "no" of the lying gnome.
If on the other hand, the truther was on the path to heaven, he would say yes. But then the other gnome is a liar so he unfaithfully report "no". So no matter which gnome you ask, the one on the road to heaven will say no. And that's a good thing isn't it?
WHAT?

OK...

Two gnomes

Only one of the gnomes is a liar

One is on the path to heaven, the other is on the path to hell.

You don't know which gnome is the liar

You don't know which path either gnome is on

So...
jimmysnyder said:
Question - Would the other gnome say he is guarding the path to heaven?

jimmysnyder said:
If the liar was on the path to heaven, he would say no.


False, he would say yes, because he's lying. Remember, you are asking the lier if the truther on the path to hell would say he is guarding the path to heaven. The truther would say no, so the lier says yes.
jimmysnyder said:
But then the other gnome is not a liar so he faithfully reports the "no" of the lying gnome.
What? The truther, when asked if the other gnome would say he's guarding the path to heaven, would say no, because the lier is on the path to heaven, so the lier would say no.

So, you have one gnome saying no and one gnome saying yes, you don't know which is lying and you don't know which path they're on. At this point you know nothing.

This isn't hard to follow guys.

You are right about how the gnomes would answer the question. Jimmy should be expecting his Vulcan death pinch any time now. :biggrin:

You are wrong that the information provided is meaningless. The way this question is phrased you can be sure the answer is a lie.

Question - Would the other gnome say he is guarding the path to heaven?
If the gnome you are asking is guarding the door to heaven,
truther = yes
liar = yes

If the gnome you are asking is guarding the door to hell,
truther = no
liar = no

yes = heaven
no = hell

edit - I think Jimmy was answering the question "Would the other gnome say you are guarding the path to heaven?" In that scenario the 'no' door would be heaven and the 'yes' door would be hell. The answer is still a lie because you are asking one gnome what the other would say.
 
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  • #38
Thank you Huck!

I was wrong about assuming the answer. Too many versions and not enough sleep, it would appear we were interpreting versions differently.

In the correct version, only one gnome would be asked and this would be the answer.

Answer: Ask if the other person would say that this is the right path?

Lets say the door is the right path
The truth person would say that the lier would say no.
The lier woud say that the truth person would say no.

Lets say the door is the wrong path.
The truth person would say the door yes.
The lier would say the truth person would say yes.

Do the opposite of what they say.

When Rogerio said the gnome would tell him yes after saying no. I took him literally to mean that the gnome verbally gave two answers "yes and no", which made no sense. What he meant was if the gnome said "no" Rogerio would know that "yes" was the correct answer. I *really* do need sleep. :frown:
 
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  • #39
This logic problem was in a Dr. Who (Tom Baker) episode.
 
  • #40
It was also in my favorite movie, 'Labyrinth.'
 
  • #41
Evo said:
...
When Rogerio said the gnome would tell him yes after saying no. I took him literally to mean that the gnome verbally gave two answers "yes and no", which made no sense. What he meant was if the gnome said "no" Rogerio would know that "yes" was the correct answer.

It is wrong again.
What I meant is clearly the opposite - read carefully my first post:

Rogerio said:
Post #16
I would just say:
"If someone asked you whether your path leads to heaven, would you answer yes?"
And the answer would be true.

It doesn't matter which gnome I'm talking to.
If the gnome says "yes" to me, then I will know that "yes" is the correct answer.
And if the gnome says "no" to me, then I will know that "no" is the correct answer.

When I ask a gnome about what he would say to someone who (hypothetically) would have asked him, then, if he is a liar, he will lie twice, and will give me verbally the correct answer. And if he is the truth gnome, of course he will tell me the correct answer, too.
 
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  • #42
This is, indeed, an oldie but a goodie. I like to explain it (when asked) by using a truth table. I hope this will help anyone who's still unsure about the solution.

You pick Gnome A and ask him, "If I asked Gnome B if his path led to heaven, would he answer 'Yes'?"

Here are the possible answers, given the possible situations:

...Gnome.A...Gnome.B
....is.the...is.the
.....liar...liar
...------------------------
..Path.A.|...|...|
leads.to.|.."Yes"...|.."Yes"...|
..heaven.|...|...|
...|----------|-----------|
..Path.B.|...|...|
leads.to.|..."No"...|..."No"...|
..heaven.|...|...|
...------------------------

The "yes" answers both correspond to the case that Path A leads to heaven, as the "no" answers correspond to Path B leading to heaven, so you know exactly what to do.

Note that there are two binary unknowns, i.e. the path to heaven (Path A or Path B) and the identity of the liar (Gnome A or Gnome B), but with only one Y/N question, you can resolve only one binary unknown, so you have to sacrifice any chance to learn who's the liar, in favor of learning which path leads to heaven. You could devise a different question that would tell you who was the liar, but you wouldn't learn which path led to heaven.
 
  • #43
jimmysnyder said:
I like the logic of Proggle's solution, and after all, it is a logic puzzle. But it does have a minor flaw. The danger is that the answer will be "I deserve to take the road to heaven.". This would not be helpful. It is easily fixed though, the logic can be preserved with a question like:


If honest gnomes go to heaven and lying ones go to hell, do you deserve to go to the left or to the right?

If you ask that.. then they could reply "the right" or "the left" you would have no clue which path leads where.. or which one is false or true
 
  • #44
Remnant said:
jimmysnyder said:
I like the logic of Proggle's solution, and after all, it is a logic puzzle. But it does have a minor flaw. The danger is that the answer will be "I deserve to take the road to heaven.". This would not be helpful. It is easily fixed though, the logic can be preserved with a question like:


If honest gnomes go to heaven and lying ones go to hell, do you deserve to go to the left or to the right?

If you ask that.. then they could reply "the right" or "the left" you would have no clue which path leads where.. or which one is false or true

No...

The honest gnome ("deserves" to go to heaven) will tell you on which side the path to heaven is. Suppose he says "I deserve to go left".

The lying gnome ("deserves" hell) would deserve the one on the right, but since he lies, he will tell you "I deserve to go left" as well.

In fact, you just need one gnome.
 
  • #45
I don't know about you guys, but I'd just look at both of them and ask:

"Gnomes, is there a fork in the road?"

One would say "yes" and one would say "no". Problem solved =).

Then I'd know which would be telling the truth and could ask "Is this the road to heaven?" to the one who answered "yes". From there depending on his answer, I'd know which road to take.
 
  • #46
DeLorean03 said:
I don't know about you guys, but I'd just look at both of them and ask:

"Gnomes, is there a fork in the road?"

One would say "yes" and one would say "no". Problem solved =).

Then I'd know which would be telling the truth and could ask "Is this the road to heaven?" to the one who answered "yes". From there depending on his answer, I'd know which road to take.
That would be two questions.
 
  • #47
You might ask "Why is one path heading up and the other heading down?"
 
  • #48
is there an answer? i doubt there is one.
it's like determining which of the statements is true and false:
The statement below is true.
The statement above is false.
 
  • #49
j0nis0n said:
is there an answer? i doubt there is one.
it's like determining which of the statements is true and false:
The statement below is true.
The statement above is false.
No, it's not like that example, which has no resolution. The puzzle in this thread does indeed have a completely logical solution, which has been stated repeatedly.
 
  • #50
After much thought I came to this conclusion

You ask each gnome what the other gnome would say if you asked them which way was heaven, the lying gnome would point to hell so that you would walk there and the gnome that tells the truth would poin to hell because that is where the liar gnome would tell you to go so you go down the other path.
 
  • #51
well, gutti, you could have saved yourself "much thought" just be reading the previous posts and seeing where this and other equivalent answers were already given. But hey - so much the better for you to have worked it out on your own!

Except that in your first case (the lying gnome), you'd walk down the other path, not the one he's pointing to (since, as you say, he's pointing to hell). That's the same choice you'd make if you'd asked the truth-telling gnome (to go down the other path), which is a good thing, since you don't know which gnome is which - and there's no way for you to tell, if you have only one question. One binary ('yes' or 'no') response can't give you two pieces of information. Luckily, you don't care which gnome is which, so long as you go down the correct road.
 
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  • #52
I didnt read the other posts till i worked it out because then there is no fun in it because I would have known the answer.
 
  • #53
gutti said:
After much thought I came to this conclusion

You ask each gnome what the other gnome would say if you asked them which way was heaven, the lying gnome would point to hell so that you would walk there and the gnome that tells the truth would poin to hell because that is where the liar gnome would tell you to go so you go down the other path.

It is not OK.
You know there is a liar and a truth teller.
But who told you they know each other?
 
  • #54
Sounds like Knights and Knaves by Raymond Smullyan!
 
  • #55
"If you knew I lied, would that make me a liar?" Heaven should say yes, hell should say no.

or

"Are you guarding a path to heaven or hell?"

or

"Is a liar a liar?"
 
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  • #56
mancerex said:
"If you knew I lied, would that make me a liar?" Heaven should say yes, hell should say no.

or

"Are you guarding a path to heaven or hell?"

or

"Is a liar a liar?"

Sorry, but it is completely wrong.
You don't know who is guarding each path.
 
  • #57
I think this is a simple riddle. Ask either guard "Which way do the likes of you go?". Either will point at the path to heaven.
 
  • #58
Werg22 said:
I think this is a simple riddle. Ask either guard "Which way do the likes of you go?". Either will point at the path to heaven.

Only if the liars always go to hell. But who knows?
If the liars can go to heaven, your question doesn't work out ok.

So, yours is not a good answer...
:smile:
 
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  • #59
Kittel Knight said:
Only if the liars always go to hell. But who knows?
If the liars can go to heaven, your question doesn't work out ok.

So, yours is not a good answer...
:smile:
No, I think it works logically. The point is that you have to ask only one Y/N question - i.e. can get only one binary answer even though there are two binary unknowns.

You could modify this suggested solution by asking, "If we were to assume, however hypothetically, that lying gnomes all go to Hell and truth-telling gnomes all go to Heaven, which path would you then follow?" The point is that you're getting one Y/N answer that unambiguously gives you the answer to one unknown (which path goes to Heaven) at the sacrifice of any information about the other one (which gnome is lying). That's the only way a problem like this can be solved, despite the many ways the question can be framed.
 
  • #60
if honest gnomes go to heaven, where would you deserve to go? the gnome telling the truth would say heaven and so will the false telling gnome because he knows to go to hell so he would say heaven.. so both gnomes would answer heaven.. this does not resolve the problem.. same with your other 2 questions.. you would get the same response from both gnomes
 

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