I have mentioned that as a young boy, I devoured Martin's Gardner's Scientific American column each month. Fork in the Road was one problem I vaguely recall he presented. (NB: On the cutting problem given earlier he did not try to obscure that multiple connected cuts would count as one cut.)

Here is a solution to the fork in the road problem without a "twist," as well as some other teasers from him.

Spoiler

https://www.theguardian.com/science...-to-martin-gardners-best-mathematical-puzzles

 

I am a little weary, but I think I have solved it.

Yes.

Would you agree that you do not have to ask any questions?

 

The first question should be....do both paths lead to heaven. That will eliminate one, possibly two, unknown bystanders.

 

I am a little weary, but I think I have solved it.
Would you agree that you do not have to ask any questions?

Actually, that answer was based on this sentence in the original puzzle:

You can choose which of the three bystanders is to answer each of your questions.

I was taught that when writing standardized test questions, "which" in that sentence can mean one, two, or all three bystanders. If the TS had meant that the traveler could ask only one bystander at a time, then he should have said "which one" of the bystanders.

Of course, the TS clarified that later, and except for posting Gardner's original question and answer, I have not given it much thought. Gardner's solution requires that responses by the bystanders be logical. Adding a random response seems to complicate it, and I am anxious to see Hymie's solution.

 

My solution is attached.

Don't click on the PDF attachment if you don't want to see it.

edited to add: I had to correct a table but the text and interpretation and solution were unchanged. I noted the edit and upload another version.

 

Actually, that answer was based on this sentence in the original puzzle:

I was taught that when writing standardized test questions, "which" in that sentence can mean one, two, or all three bystanders. If the TS had meant that the traveler could ask only one bystander at a time, then he should have said "which one" of the bystanders.

Of course, the TS clarified that later, and except for posting Gardner's original question and answer, I have not given it much thought. Gardner's solution requires that responses by the bystanders be logical. Adding a random response seems to complicate it, and I am anxious to see Hymie's solution.

Yes, I agree that it is essential that you be extremely clear when asking the question. I tried to do that here.
I will only be a little embarrassed if I screwed up the answer that I already posted, but I do think I solved the puzzle (this time). I am not going to go back and check (at least not right now), but my thinking is that the answer depends on the questions asked (this will make sense if you read the last line in my "solution". That is the effect of the random bystander. If I am correct in my solution and the effect of the random bystander, I can only say that I have not come across such a "twist" before and, again, if I am correct, the set up for the problem may need to be explained a little better. But, maybe I just missed something fundamental in my "solution" and it is no solution at all.

 

Looking at your complex analysis in ‘RoadToHeavenA’ you appear to have determined who each of the bystanders are (based on their responses to the question), where you know (assume) which road leads to heaven. But given that you don’t know which road leads to heaven, you cannot say which of the two sets of analysis is correct and therefore do not know which is the road to heaven.

As an example, you ask B1 & B2 if Road 1 leads to heaven (and there is a difference of opinion).
B1 can be L/T/R, B2 can be L/T/R (obviously both cannot be the same); this leaves B3 able to give any response (to the question) because you don’t know if Road 1 leads to heaven.

If you want to test your theory that you have solved the problem, I will act as the three bystanders and give responses to your questions; after which you should be able to announce which road leads to heaven.

 

Looking at your complex analysis in ‘RoadToHeavenA’ you appear to have determined who each of the bystanders are (based on their responses to the question), where you know (assume) which road leads to heaven. But given that you don’t know which road leads to heaven, you cannot say which of the two sets of analysis is correct and therefore do not know which is the road to heaven.

As an example, you ask B1 & B2 if Road 1 leads to heaven (and there is a difference of opinion).
B1 can be L/T/R, B2 can be L/T/R (obviously both cannot be the same); this leaves B3 able to give any response (to the question) because you don’t know if Road 1 leads to heaven.

If you want to test your theory that you have solved the problem, I will act as the three bystanders and give responses to your questions; after which you should be able to announce which road leads to heaven.

I do not agree with you at all. It could be that I am misunderstanding what you are saying. The analysis I presented determined the road to heaven.


When you say, "As an example, you ask B1 & B2 if Road 1 leads to heaven (and there is a difference of opinion)."

That is clearly covered by the two cases where the answer sequence for B1 and B2 is YN and NY.

"this leaves B3 able to give any response (to the question) because you don’t know if Road 1 leads to heaven." That is why I continue to run the analysis of B3 given each possible identity for B3 R/T/L- and for R1=H and R2=H - I don't think that you see that, but I did it very clearly.

Now, I have no problem admitting that this is not a solution, if you (or anyone) show me how it is not a solution. I don't believe that you have done so yet.

Further, I wrote seven pages of explanation - now I think it is time for you to put up a spoiler solution of your own.

Edited to add: I absolutely did not predetermine which road leads to heaven, but solved it without any such presumption. Carefully re-read the analysis and when you do, take note of the last sentence.

 

Here's another idea.
Based on the solution to the original problem (without the twist) that you can gain information by asking one of the two participants what the other would say, guaranteeing a lie.

If you ask "if you ask the other two 'is R1 the road to heaven?' will they agree that it is 50% of the time?" then T (truth teller) and L (liar) will both tell you NO if R1 is the road to heaven and YES if it isn't, whereas R (random) will be random. So you can answer the problem by asking this question to all three and take the majority (or possibly consensus). I can't see how to do it with two questions.

 

...

If you want to test your theory that you have solved the problem, I will act as the three bystanders and give responses to your questions; after which you should be able to announce which road leads to heaven.

Sure, I could go for that, of course you would need to send the identities of the bystanders (B1, B2, B3 as T, L, or R, one of each) and which road goes to heaven (R1 or R2) to a third party here (a neutral user), who will not contact me until after I have posted a solution following three questions. This way, we all know that nothing changes. If I provide the correct solution you will post that it "is not a theory, it is a solution". If I am incorrect, I will post, "it is not only a theory, it is a bad one".

Let me know when you have sent it to someone and who it is and then you can answer the questions and after three questions, I will reveal the road to heaven. Edited to add: Also, I am not just going to guess a road, I will explain the reasoning as the answers to the questions are posted by you.

My first two questions are:

Q1 is to B1 - Is R1 the road to heaven?
Q2 is to B2 - is R1 the road to heaven?

________________
I also want to say that I have already seen a couple of errors in my solution document - nothing really big (e.g., I have written "but in each case R1=H" when it should obviously have said R2=H and there are clearly solutions that yield R2=H). I will correct them and repost it with a revised name when I have a chance and can take the time to check it carefully. But, that does not change anything for the challenge.

The solution strategy remains:
Ask Q1 to B1
Ask Q2 to B2 (Q1 and Q2 are the same question, e.g., Is R1 the road to heaven?)

Then, Q3 depends on the sequence of answers. YY NN YN or NY. In the case of YY and NN, Q3 is to ask B3 the same question that was asked to B1 and B2. In the case of YN and NY, Q3 doesn't really matter, you decide based upon an analysis of all of the possible identities of
B3.

Agreed?

 

In answer to your question (is R1 the road to heaven?):-

B1 responds – Yes

B2 responds – No

Now ask B3 your question, I will respond with the answer – and you can then tell me which road leads to heaven.

 

In answer to your question (is R1 the road to heaven?):-

B1 responds – Yes

B2 responds – No

Now ask B3 your question, I will respond with the answer – and you can then tell me which road leads to heaven.

Sure, I could go for that, of course you would need to send the identities of the bystanders (B1, B2, B3 as T, L, or R, one of each) and which road goes to heaven (R1 or R2) to a third party here (a neutral user), who will not contact me until after I have posted a solution following three questions. This way, we all know that nothing changes. If I provide the correct solution you will post that it "is not a theory, it is a solution". If I am incorrect, I will post, "it is not only a theory, it is a bad one".

Let me know when you have sent it to someone and who it is and then you can answer the questions and after three questions, I will reveal the road to heaven. Edited to add: Also, I am not just going to guess a road, I will explain the reasoning as the answers to the questions are posted by you.

 

Here's another idea.
Based on the solution to the original problem (without the twist) that you can gain information by asking one of the two participants what the other would say, guaranteeing a lie.

If you ask "if you ask the other two 'is R1 the road to heaven?' will they agree that it is 50% of the time?" then T (truth teller) and L (liar) will both tell you NO if R1 is the road to heaven and YES if it isn't, whereas R (random) will be random. So you can answer the problem by asking this question to all three and take the majority (or possibly consensus). I can't see how to do it with two questions.

Whilst I completely agree with your logic, the problem with the question you ask is that all three could answer No (if random answers No).

In solving this puzzle you must assume that the random bystander gives the response that is most unhelpful to your line of questioning – yet when the random bystander answers so, you can still determine which road leads to heaven.

 

Whilst I completely agree with your logic, the problem with the question you ask is that all three could answer No (if random answers No).

In solving this puzzle you must assume that the random bystander gives the response that is most unhelpful to your line of questioning – yet when the random bystander answers so, you can still determine which road leads to heaven.

I think using Gardner's logical question, you only need 2 out of the 3 answers to correspond (i.e., the liar's response is always the same as the truth teller's). My interpretation of his logic is that it converts a "liar" to a truth teller, much like using negative logic gates to give an "and." More generally, negative logic allows "and" and not; positive logic does not.

 

Whilst I completely agree with your logic, the problem with the question you ask is that all three could answer No (if random answers No).

In solving this puzzle you must assume that the random bystander gives the response that is most unhelpful to your line of questioning – yet when the random bystander answers so, you can still determine which road leads to heaven.

It doesn't matter what answer the random bystander gives.

Ask all three
"if you ask your two colleagues: 'is R1 the road to heaven?' is it possible that they will both say YES?"

1. R1 is the road to heaven:
you will get either NNY or NNN (random guy last)

2. R1 isn't the road to heaven:
you will get either YYY or YYN

as these are distinct, these three questions solve the problem.

In fact, if you ask the questions sequentially, half the time you can stop after the second question, as as soon as you get two answers that agree you know which road leads to heaven. This works even if one of the two answers comes from the random bystander. So this solves the problem with 2.5 questions!

 

I think using Gardner's logical question, you only need 2 out of the 3 answers to correspond (i.e., the liar's response is always the same as the truth teller's). My interpretation of his logic is that it converts a "liar" to a truth teller, much like using negative logic gates to give an "and." More generally, negative logic allows "and" and not; positive logic does not.

If that is true then the three questions to B1, B2 and B3, are all simply, "If I were to ask you if R1 goes to heaven, would you say yes?" The question being asked in such a way that L must tell the truth about the road while answering the question with a lie. This is straight from the Gardner link that you presented (and a bazillion other places).

If R1=H
L says Yes
T says Yes
R says either Y/N

If R2=H
L says No
T says No
R says either Y/N

The idea is that 2 or 3 Yes answers mean R1=H and 2 or 3 No answers mean R2=H.

If that is the solution it is a simple extension of Gardner's and is is distinctly disappointing.

 

It doesn't matter what answer the random bystander gives.

Ask all three
"if you ask your two colleagues: 'is R1 the road to heaven?' is it possible that they will both say YES?"

1. R1 is the road to heaven:
you will get either NNY or NNN (random guy last)

2. R1 isn't the road to heaven:
you will get either YYY or YYN

as these are distinct, these three questions solve the problem.

In fact, if you ask the questions sequentially, half the time you can stop after the second question, as as soon as you get two answers that agree you know which road leads to heaven. This works even if one of the two answers comes from the random bystander. So this solves the problem with 2.5 questions!

This is the correct answer, once two of the answers agree the solution is known (even if one of the respondents is the random bystander). But the original question was what is the minimum number of questions you need ask to guarantee that you know which road leads to heaven?

 

Just looking through some old Larsen cartoons and couldn't resist:

 

How about this question:

If I ask you "Is this the road to heaven, would your answer be yes?"

Majority vote wins.

 

Sure, I could go for that, of course you would need to send the identities of the bystanders (B1, B2, B3 as T, L, or R, one of each) and which road goes to heaven (R1 or R2) to a third party here (a neutral user), who will not contact me until after I have posted a solution following three questions. This way, we all know that nothing changes. If I provide the correct solution you will post that it "is not a theory, it is a solution". If I am incorrect, I will post, "it is not only a theory, it is a bad one".

Let me know when you have sent it to someone and who it is and then you can answer the questions and after three questions, I will reveal the road to heaven. Edited to add: Also, I am not just going to guess a road, I will explain the reasoning as the answers to the questions are posted by you.

Why does he need to send anything to a third party? What does it matter if he changes things midstream? Once your interaction is complete you either have a unique, determinitistic solution that depends ONLY on the questions asked, who they are asked of, and the answers given, or you don't. Hymie is free to change whatever he wants, as long as his final choice of assignments are consistent with the answers provided to the questions asked.

In essence, he starts out with every possible path through the decision tree available to him and, at each step he gets to choose his answers so that so as to prune as few of those paths as possible (or, more to the point, to keep paths that give him the most flexibility).

At the end of the day, as long as he can show that there is at least one assignment of roads and roles that results in you getting the answers he provided to the questions you asked while also resulting in you picking the wrong road, then he has successfully demonstrated that your algorithm is flawed.

Note that the reverse is NOT the case -- if Hymie can't come up with an assignment that makes your solution incorrect for the questions asked and answers provided, that does NOT prove that your algorithm is correct, only that it is capable, at least under some circumstances, of providing a correct result. But your algorithm is a solution to the problem only if it provides a correct result under ALL circumstances.