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The road to heaven ...

UncleEbenezer
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The road to heaven ...

#477133

Postby UncleEbenezer » January 29th, 2022, 6:14 pm

This board's been a bit quiet, so here's something I encountered way back before I'd even embarked on the maths/science trajectory of my O-levels. I expect most of us know an oft-repeated simpler version of this, but sad news here brought it to mind.

On the road to heaven[1] are two unmarked gates: one leads to heaven, the other to hell. They are of course indistinguishable until after you've irrevocably passed.

At the gate are three shadowy figures, seated on thrones labelled A, B and C. In this place, if not in life, they too are indistinguishable. They will answer two yes/no questions before vanishing. You can ask both to the same figure, or one to each of two of them.

The Simpleton, S, will always answer truthfully, for he can do no other.
The QC, Q, always lies, from a distinguished career representing Clients whose enemy is truth.
The Prime Minister, P, says whatever he considers expedient without regard to truth or lies.

How do you ask your two questions?

The Good News is that Gengulphus found it easier than any of us.



[1] Um, for those who solve the puzzle, obviously.

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Re: The road to heaven ...

#477444

Postby jfgw » January 31st, 2022, 10:03 am

Do the shadowy figures know each other's professions? I have an answer if they do.


Julian F. G. W.

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Re: The road to heaven ...

#477448

Postby UncleEbenezer » January 31st, 2022, 10:26 am

jfgw wrote:Do the shadowy figures know each other's professions? I have an answer if they do.


Julian F. G. W.

Hehe.

When I first encountered this, the figures were named: Gandhi (truth), Goebbels (lies) and de Gaulle (unpredictable). I think that kind-of dates my source, but was already a bit outdated, and my (I think) pre-teen self at the time didn't know who they were.

The names - or indeed the updated professions - are of course immaterial. But I'm sure your answer has amusement value. :D

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Re: The road to heaven ...

#477455

Postby jfgw » January 31st, 2022, 10:54 am

UncleEbenezer wrote:The names - or indeed the updated professions - are of course immaterial. But I'm sure your answer has amusement value.


I'm not sure about amusement value but I am trying to word it without an IF...THEN...ELSE statement.


Julian F. G. W.

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Re: The road to heaven ...

#477473

Postby SteelCamel » January 31st, 2022, 11:39 am

As far as I can see, the puzzle is only solvable if:
All three figures are aware of which gate leads where
All three figures know the identity and behaviour of each figure (including themselves)

Without these assumptions, there's no guarantee of getting a useful answer no matter what you ask.

With them, there is a reasonably simple solution. The key is to realise that with the appropriate questions you can know which gate to enter even though you still haven't identified any of the three figures.

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Re: The road to heaven ...

#477509

Postby UncleEbenezer » January 31st, 2022, 1:17 pm

SteelCamel wrote:As far as I can see, the puzzle is only solvable if:
All three figures are aware of which gate leads where
All three figures know the identity and behaviour of each figure (including themselves)

Ah, yes. This is indeed an appropriate place for pedantry. I guess that was Julian's point, too. Go ahead.

Out of interest, is this a problem you've seen? The one that seems to come up regularly is a simpler version with just two figures.

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Re: The road to heaven ...

#477553

Postby 9873210 » January 31st, 2022, 2:50 pm

jfgw wrote:
UncleEbenezer wrote:The names - or indeed the updated professions - are of course immaterial. But I'm sure your answer has amusement value.


I'm not sure about amusement value but I am trying to word it without an IF...THEN...ELSE statement.


Julian F. G. W.


Have you tried using XOR? Although that might confuse simpleton and give QC grounds for appeal.

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Re: The road to heaven ...

#477562

Postby jfgw » January 31st, 2022, 3:18 pm

Spoiler

This is not a puzzle that I have seen before.

Step one is to eliminate the politician.

I shall refer to the shadowy figures by the letters of the thrones upon which they sit.

First, I would ask A the following:

"If B is the politician and I asked C, or if B is the simpleton or the QC and I asked B this question: 'is B the politician?', would the answer be "yes"?"

If B is the politician, the answer would be "no", and if C is the politician, the answer would be "yes".

If A is the politician, the answer is undefined but we know that neither B, nor C can be the politician.

Therefore, if the answer is "no", we know that C cannot be the politician, and if the answer is "yes", we know that B cannot be the politician. The second question can, therefore, be directed to a shadowy figure that we can be certain is not the politician.

The second question is a bit simpler:

"If I asked the other shadowy figure who is not the politician if Heaven's door is on the left, what would he say?"

If heaven's door is on the left, the answer would be "no", and if heaven's door is on the right, the answer would be "yes".



Julian F. G. W.

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Re: The road to heaven ...

#477579

Postby mc2fool » January 31st, 2022, 4:01 pm

UncleEbenezer wrote:
SteelCamel wrote:As far as I can see, the puzzle is only solvable if:
All three figures are aware of which gate leads where
All three figures know the identity and behaviour of each figure (including themselves)

Ah, yes. This is indeed an appropriate place for pedantry. I guess that was Julian's point, too. Go ahead.

Out of interest, is this a problem you've seen? The one that seems to come up regularly is a simpler version with just two figures.

Indeed, the two figure version, one which always tells the truth and one which always lies, and you only get to ask one question of one of them, is the version I'm familiar with.

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Re: The road to heaven ...

#477584

Postby UncleEbenezer » January 31st, 2022, 4:23 pm

jfgw wrote:Spoiler
First, I would ask A the following:

My mind boggles at your first question :? .

But it seems to work, and you got straight to the crux of the problem. Congratulations. 8-)

But your second question really shouldn't be allowed: I must've formulated the question too carelessly to permit (or appear to permit) that formulation. I should have specified, you can only refer to the figures as A, B and C in your questions. The first question (for all its complexity) doesn't make any assumption you couldn't reformulate that way, but the second one I think does.

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Re: The road to heaven ...

#477598

Postby 9873210 » January 31st, 2022, 4:54 pm

UncleEbenezer wrote:
jfgw wrote:Spoiler
First, I would ask A the following:

My mind boggles at your first question :? .

But it seems to work, and you got straight to the crux of the problem. Congratulations. 8-)

But your second question really shouldn't be allowed: I must've formulated the question too carelessly to permit (or appear to permit) that formulation. I should have specified, you can only refer to the figures as A, B and C in your questions. The first question (for all its complexity) doesn't make any assumption you couldn't reformulate that way, but the second one I think does.

You can reformulate "If I ask the other figure who is not a politician ... " as "If A is not the politician and I ask A ... or If B is not the politician I ask B ... ". Same thing, just more verbose.


But for the two person problem I have always prefered.
"If I asked you does the left door lead to heaven would you say yes" over If I asked the other one door leads to heaven would you say yes." As far as I can see these have the same level of complexity, but many people think the first is too recursive.

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Re: The road to heaven ...

#477600

Postby UncleEbenezer » January 31st, 2022, 5:02 pm

9873210 wrote:
UncleEbenezer wrote:
jfgw wrote:Spoiler
First, I would ask A the following:

My mind boggles at your first question :? .

But it seems to work, and you got straight to the crux of the problem. Congratulations. 8-)

But your second question really shouldn't be allowed: I must've formulated the question too carelessly to permit (or appear to permit) that formulation. I should have specified, you can only refer to the figures as A, B and C in your questions. The first question (for all its complexity) doesn't make any assumption you couldn't reformulate that way, but the second one I think does.

You can reformulate "If I ask the other figure who is not a politician ... " as "If A is not the politician and I ask A ... or If B is not the politician I ask B ... ". Same thing, just more verbose.

Same thing, same issue. Your "if" clause ascribes identity, which I never meant to allow.

But your preferred question in the spoiler is an ideal second question.

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Re: The road to heaven ...

#477607

Postby 9873210 » January 31st, 2022, 5:40 pm

UncleEbenezer wrote:
9873210 wrote:
UncleEbenezer wrote:
jfgw wrote:Spoiler
First, I would ask A the following:

My mind boggles at your first question :? .

But it seems to work, and you got straight to the crux of the problem. Congratulations. 8-)

But your second question really shouldn't be allowed: I must've formulated the question too carelessly to permit (or appear to permit) that formulation. I should have specified, you can only refer to the figures as A, B and C in your questions. The first question (for all its complexity) doesn't make any assumption you couldn't reformulate that way, but the second one I think does.

You can reformulate "If I ask the other figure who is not a politician ... " as "If A is not the politician and I ask A ... or If B is not the politician I ask B ... ". Same thing, just more verbose.

Same thing, same issue. Your "if" clause ascribes identity, which I never meant to allow.


It's the same formulation as Julean's first question, which you either allowed or believe can be restated in a way you would allow. Not clear what the rules are at this point.

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Re: The road to heaven ...

#477610

Postby jfgw » January 31st, 2022, 5:45 pm

UncleEbenezer wrote:But your second question really shouldn't be allowed: I must've formulated the question too carelessly to permit (or appear to permit) that formulation. I should have specified, you can only refer to the figures as A, B and C in your questions. The first question (for all its complexity) doesn't make any assumption you couldn't reformulate that way, but the second one I think does.


If the first question dictated that the second question should be addressed to B, I could ask, "If I asked A and C if Heaven's door is on the left, is it possible that both answers would be "yes"?, and
If the first question dictated that the second question should be addressed to C, I could ask, "If I asked A and B if Heaven's door is on the left, is it possible that both answers would be "yes"?

(I think that there's an AND in each of those somewhere, I'm not sure how I would use an XOR.)


Julian F. G. W.

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Re: The road to heaven ...

#477612

Postby jfgw » January 31st, 2022, 5:52 pm

9873210 wrote:It's the same formulation as Julean's first question, which you either allowed or believe can be restated in a way you would allow. Not clear what the rules are at this point.


The rules may depend upon the answer to the first reply,
jfgw wrote:Do the shadowy figures know each other's professions? I have an answer if they do.



Julian F. G. W.

(Waiting for a pedant to point out that "simpleton" is not a profession.)

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Re: The road to heaven ...

#477624

Postby 9873210 » January 31st, 2022, 7:06 pm

jfgw wrote:If the first question dictated that the second question should be addressed to B, I could ask, "If I asked A and C if Heaven's door is on the left, is it possible that both answers would be "yes"?, and

One of A or C is P. We (and S and Q) know that P answers to his advantage. We may or may not know what is to his advantage. P may always flip a coin (to save having to make a decision). P may always answer "no" (because he has to pay by the letter). P may always answer "yes" (because he is paid by the letter). In any of these case S or Q may or may not know this. Different states of knowledge give different answers.

If P is a Byzantine General, and it is to his advantage to send you to one way or the other, and S and Q know this, your question may collapse the universe into a paradox before you get an answer.

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Re: The road to heaven ...

#477631

Postby UncleEbenezer » January 31st, 2022, 8:02 pm

9873210 wrote:It's the same formulation as Julean's first question, which you either allowed or believe can be restated in a way you would allow. Not clear what the rules are at this point.

I said you shouldn't ascribe identity.

You can of course ask any of the figures what he or any other of the figures would say in reply to a hypothetical question. It is not necessary at any point to ask the kind of thing Julian posted "if B is P ..." that ascribes hypothetical identity to the figures.

You can get round that by asking for the characteristics of the identity: if B is P then he is more likely to tell the truth than Q, but less likely than S. That's the basis on which I feel Julian's first question could be allowed.

The basis for allowing his second question is that the problem was underspecified (hey, now I have an idea how cinelli feels when I do that :roll: ). Also on that basis, I'm trying to shift the goalposts to what I originally meant.

Let's put it like this. A knows whether B is P by virtue of knowing exactly how likely B is to tell the truth. But "Prime Minister" or "Politician" isn't in his vocabulary, so he can't be asked that question. I'd to say, neither are proxy descriptions like "truth-teller", "liar", but that could be read as making the problem impossible.

I guess what I dislike but can't really disallow are conditional clauses like "if B is ...". Perhaps I can play the Speaker and rule them Unparliamentary? After all, Honourable Members aren't allowed to call each other Liars! :D

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Re: The road to heaven ...

#477667

Postby 1nvest » February 1st, 2022, 12:01 am

With a QC, PM and simpleton guarding the gates, think I'd be inclined to not even bother asking any questions, turn around and return to hell.

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Re: The road to heaven ...

#477727

Postby SteelCamel » February 1st, 2022, 10:44 am

UncleEbenezer wrote:Out of interest, is this a problem you've seen? The one that seems to come up regularly is a simpler version with just two figures.


Yes, I have seen something similar, which is why I didn't jump straight in with an answer.
The simpler version is more well known. If you ask "If I asked you whether the left gate leads to heaven, would you say yes?" then both S and Q will give you the correct answer - S will answer honestly, Q will lie but will then lie about lying and so inadvertently give the correct answer. So that's your second question. Note that it doesn't matter whether it was addressed to S or Q, and you don't need to know who you're addressing - though you must be sure you're not asking P as they can give you the wrong answer.

So your first question must identify someone as not being P. It doesn't matter who they are, only who they are not. The trick is to ask the first question to a different person. If you ask your first question to P, the answer will be useless - but it doesn't matter, as either of the other two will give you the answer you need. If you ask your first question to either S or Q, you can locate P and avoid them. So pick one, let's say A. Ask them "If I asked you whether the figure on throne B is P would you say yes?". By the same reasoning as above, S or Q will give you the correct answer - if they say yes, B is P, so ask the question to C, if not then B is not P so ask it to B. And if A was P, it doesn't matter whether you ask B or C.

So step by step:
Ask A: "If I asked you whether the figure on throne B is P, would you say yes?"
If they say yes, address your next question to C, if they say no address your next question to B.
Ask your choice of B or C: "If I asked you whether the left gate leads to heaven, would you say yes?"
If they say yes, take the left gate. If they say no, take the right one.

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Re: The road to heaven ...

#477751

Postby jfgw » February 1st, 2022, 11:35 am

SteelCamel wrote:Ask A: "If I asked you whether the figure on throne B is P, would you say yes?"
If they say yes, address your next question to C, if they say no address your next question to B.
Ask your choice of B or C: "If I asked you whether the left gate leads to heaven, would you say yes?"
If they say yes, take the left gate. If they say no, take the right one.

Admittedly, that answer does look a bit simpler than mine :)


Julian F. G. W.


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