game theory / logic puzzle

[NotJeff]

Originally Posted by Ox

I believe I might have solved the truth machines.

Call the machines A, B, C. Ask A, "Is it true that B works if and only if you use green to mean true?"

Ox has it. The central insights that reduces this to the more standard knight/knave problem is "simply" that

you need to use one machine to help you decide between the other two, so that if the one machine is crazy it's fine no matter which other you pick

.

[TheKeck]

Originally Posted by NotJeff

Ox has it. The central insights that reduces this to the more standard knight/knave problem is "simply" that

you need to use one machine to help you decide between the other two, so that if the one machine is crazy it's fine no matter which other you pick

.

The other key things I was missing were:

Basically encoding multiple questions into a single question.

Making part of the question about what the colors mean.

[NotJeff]

Originally Posted by TheKeck

The other key things I was missing were:

Basically encoding multiple questions into a single question.

Making part of the question about what the colors mean.

Well, yes, but that's part of the "standard" knave/knight puzzle:

You have before you a knight and a knave. One ALWAYS tells the truth, and the other ALWAYS lies. Ask one of them one question, and determine which is which.

There are two solutions, both of which involve your "key things" above:

Option 1:

If I asked you if you were a knight, what would your answer be?

The knight will answer yes, the knave will answer no. You've forced the knave to be truthful by asking him a convolution of two questions, getting the truth-about-the-truth from the knight or a lie-about-a-lie (which in this two-state case is truth) from the knave

Option 2:

If I asked the other guy if you were a knight, what would he say?

The knight will answer "no", and the knave will answer "yes". You flip the answer to get the reality. In this case the knight is telling the truth about a lie, and the knave is telling a lie about the truth. In each case you know you are negating exactly one falsehood.

[Ink Asylum]

You've nested spoilers. At least for me I can't open the nested one.

[NotJeff]

Originally Posted by Ink Asylum

You've nested spoilers. At least for me I can't open the nested one.

Dang. That works in some forums. Here it is unrolled:

You have before you a knight and a knave. One ALWAYS tells the truth, and the other ALWAYS lies. Ask one of them one question, and determine which is which.

There are two solutions, both of which involve your "key things" above:

Option 1:

If I asked you if you were a knight, what would your answer be?

The knight will answer yes, the knave will answer no. You've forced the knave to be truthful by asking him a convolution of two questions, getting the truth-about-the-truth from the knight or a lie-about-a-lie (which in this two-state case is truth) from the knave

Option 2:

If I asked the other guy if you were a knight, what would he say?

The knight will answer "no", and the knave will answer "yes". You flip the answer to get the reality. In this case the knight is telling the truth about a lie, and the knave is telling a lie about the truth. In each case you know you are negating exactly one falsehood.

[TheKeck]

I guess I'm just not seeing how that employs either key I outlined.

The solution from Labyrinth and the one I always use is asking what the other guys would answer. Because you know you are getting basically one truth and one lie, you know to flip the answer.

But, you are still just asking one fairly straightforward question. This strikes me as very dissimilar to positing two predicates and then asking if exactly one of the two is true. Also, true and false are still well established. Part of the question isn't about "what does true mean?".

[NotJeff]

Originally Posted by TheKeck

I guess I'm just not seeing how that employs either key I outlined.

TheKeck Key #2:

Making part of the question about what the colors mean.

Applicability:

When talking to knights and knaves you don't know what yes and no mean. So you have to ask a question such that it doesn't MATTER. That's approximately the same trick as in the Truth Machine question: ask a question such that it doesn't matter what yes/no/plaid/paisley mean.

TheKeck Key #1:

Basically encoding multiple questions into a single question.

Applicability:

This is the trick you use to MAKE color meaning not matter. In both the basal knight/knave and in Truth Machine problem, you do it by crafting nested or multipart questions into one such that you force a particular set of possible meanings that come from different sources but drive the same result behavior to the questioner onto the same answer.

Re: Labyrinth:

What would you say about what the other guy would say is maybe more straightforward than the Truth machine answer, but it's also definitely a compound question.

[TheKeck]

I've been thinking about this way too much.

Another thing about the Labyrinth problem is that you aren't trying to determine who is the knight and who is the knave. You are trying to find out which door to take. The answer depends on being able to get reliable answers from one knight and one knave. After asking your question, you are left knowing no more about who is a knight and who is a knave than when you started.


I'm starting to see how classic the knight/knave problem is a bit more applicable, though. First of all, it doesn't depend on knowing you have a knight/knave combo. You can simply ask the question to any given person to determine if they are a knight or knave. (Incidentally, this scenario also has trivially easy solutions like "Is the sky blue?") Thinking about it this way, though, the green vs red color thing basically becomes the knight vs knave question. I was thinking of the working machines as knights and the broken machine as an ALMOST knave. I should have been thinking of a machine which uses green for true as a knight and a machine that uses red for true as a knave. I guess that's one connection other people saw easily and just went right over my head.

[NotJeff]

Originally Posted by TheKeck

I've been thinking about this way too much.

Another thing about the Labyrinth problem is that you aren't trying to determine who is the knight and who is the knave. You are trying to find out which door to take. The answer depends on being able to get reliable answers from one knight and one knave. After asking your question, you are left knowing no more about who is a knight and who is a knave than when you started.


I'm starting to see how classic the knight/knave problem is a bit more applicable, though. First of all, it doesn't depend on knowing you have a knight/knave combo. You can simply ask the question to any given person to determine if they are a knight or knave. (Incidentally, this scenario also has trivially easy solutions like "Is the sky blue?") Thinking about it this way, though, the green vs red color thing basically becomes the knight vs knave question. I was thinking of the working machines as knights and the broken machine as an ALMOST knave. I should have been thinking of a machine which uses green for true as a knight and a machine that uses red for true as a knave. I guess that's one connection other people saw easily and just went right over my head.

You're right, I bungled knight/knave a bit. The classic problem usually requires not figuring out who's knight and knave but answering some otherwise unknowable question. "Which door has the tiger behind it" or "Which road leads to freedom" or whatever and in the end DOESN'T depend on knowing which dude is which.

[TheKeck]

Here's a question that's always puzzled me. Why would someone define them self by what their name is NOT?

[NotJeff]

Originally Posted by TheKeck

Here's a question that's always puzzled me. Why would someone define them self by what their name is NOT?

I'm willing to give hints by PM. But you don't want that, do you?

[TheFlyingOrc]

Originally Posted by Reelya

Here's a fun one:

There are two types of inhabitants in the world: humans and vampires. Humans always tell the truth and vampires always lie. Also, inhabitants can either be sane or insane. Someone who is insane has all of their beliefs backwards; everything true they believe to be false, and everything false they believe to be true.

For example, if you ask an insane vampire, “Is the sky blue?” he will insanely think that it isn’t and then lie and say that it is. He will answer, “Yes.”

What is a single yes or no question that you could ask to any inhabitant to determine if he is a human or a vampire?

"Did you think you could fool me, filthy vampire? Are you enjoying the stake that I've driven into your heart? Did you think that I didn't know that I'M THE ONLY REAL PERSON LEFT?"

[TheKeck]

Originally Posted by TheFlyingOrc

"Did you think you could fool me, filthy vampire? "

I'm trying to think about how an insane vampire would answer this, and it's sort of hurting my head.

It probably doesn't help to ask subjective questions in these sort of puzzles.

[Panthera]

Not unless you want to get bayesian.

[NotJeff]

Originally Posted by Panthera

Not unless you want to get bayesian.

There are two types of people on these boards: those that have an 87% chance of understanding that comment, and those that have a 16% chance.