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# Truth, lies and pot luck

## Question

Monty Hall has two doors, concealing in some order a valuable prize and a bag of garbage. But when Regis Philbin learns that the prize is \$1 million, he runs onto stage and has security throw Monty out, claiming Monty has stolen his show. Regis then asks if you want to be a millionaire. Since you haven't answered any questions so far and thus have nothing to lose, you agree. He tells you that the contents behind the door of your choice are yours to keep. Which door do you choose? As an unadvertised bonus, you may use one of the standard lifelines for assistance.

You reason that since none of your friends has any useful knowledge, a phone call is useless. Accordingly, you ask the computer to eliminate half of the answers. Unfortunately the computer has been programmed to eliminate precisely two answers. This is useful when there are four choices. But with only two options, it also is not useful. That leaves the audience call-out.

The good news is that the audience was shown the winning door before you arrived. Every person in the audience has the information that you need to win the money. The bad news is that the audience comprises unknown numbers of truth-tellers, liars, and persons who randomly tell the truth or lie.

Regis senses your sadness and offers you a choice. You may ask a single yes/no question of your choosing, either to the audience as a whole, after which you will see a tabulation of their answers, or to a single audience member, whom you may select by pointing out, from where you sit on stage.

I know. This plot is worn pretty thin. But if something isn't clear, ask..

Otherwise, just tell me how much money you go home with, and how. Enjoy!

## Recommended Posts

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the audience is a red herring. Assuming a single audience member who can be truthteller, liar or undecided/human

, a single question that uses the truth value of the answer inside the question might do the trick for going away with the whole prize all the time. Higher order logic though, bu then so are the axioms about the three types of people.

Truth teller says yes/no equivalent to prize being behind door 1/ being before door 2

Liar would give opposite answer to what he would normally evaluate the expression. If a and b then c.

If prize is behind door one and he would lie his answer would be no. So trying to lie about the outcome pf the hypothetical question, he says yes. Similar argument for the other case.

Undecided person is forced by the paradox of the question to evaluate the trith value of the actual answer before evaluating the answer itself. A machine might get syntactically stuck but a person shoul be able to evaluate/decide the truth value / context before evaluating the question itself. Therefore after he/she does that, the evaluation goes a deterministic path. Hence his/her answer would match the same equivalence as the other types discussed above.

Key here is if this is allowed by the OP.

The wording I intepreted would suggest this apparent paradox would qualify.

Then again, I've already decided if I'm gonna lie or tell the truth in this post

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Ask the entire audience this question "If I asked you individually whether the prize is behind the left door what would your answer be?"

Both truth-tellers and liars will answer the same as the liars are forced to lie twice and will provide the answer I need. The random answerers are the culprit here as they still can answer truthfully or lie. If the population of the audience is more or less evenly distributed then I will go home with \$1MM by following the majority. If the audience is heavily populated with the randoms then there is a chance that the majority vote will point to the wrong door.

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Ask the entire audience this question "If I asked you individually whether the prize is behind the left door what would your answer be?"

Both truth-tellers and liars will answer the same as the liars are forced to lie twice and will provide the answer I need. The random answerers are the culprit here as they still can answer truthfully or lie. If the population of the audience is more or less evenly distributed then I will go home with \$1MM by following the majority. If the audience is heavily populated with the randoms then there is a chance that the majority vote will point to the wrong door.

Nice.

That would get you three points of the star.

But there is a more complete solution.

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What kind of 'random' are the random answerers? Do they, like, flip the coin every time they answer or is there a pre-set random order of truth/lie that they have memorized? Also what happens when you ask someone a question they cannot answer?

...would be something like: Ask the entire audience "if I asked you N questions, then for the (N+1)th question asked you 'Is the money in the left door?', would you answer yes?"

Truth tellers and liars will both give you the truth, random answerers will not be able to answer.
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My understanding of the "random answerer" is that he flips a coin every time you ask any question and answers "Yes" for heads and "No" for tails. It doesn't really matter what the question is as the answer will be either a truth or a lie, provided the question eliminates any other kind of answer as valid (yes/no question). There isn't a yes/no question they cannot answer.

Edited by k-man
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What kind of 'random' are the random answerers? Do they, like, flip the coin every time they answer or is there a pre-set random order of truth/lie that they have memorized? Also what happens when you ask someone a question they cannot answer?

...would be something like: Ask the entire audience "if I asked you N questions, then for the (N+1)th question asked you 'Is the money in the left door?', would you answer yes?"Truth tellers and liars will both give you the truth, random answerers will not be able to answer.

As stated in the OP any audience member who is not a truth-teller or a liar will randomly lie or tell the truth. If you like, s/he could make that decision by coin flip.

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My understanding of the "random answerer" is that he flips a coin every time you ask any question and answers "Yes" for heads and "No" for tails. It doesn't really matter what the question is as the answer will be either a truth or a lie, provided the question eliminates any other kind of answer as valid (yes/no question).

The OP does not describe them as randomly answering yes or no.

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What kind of 'random' are the random answerers? Do they, like, flip the coin every time they answer or is there a pre-set random order of truth/lie that they have memorized? Also what happens when you ask someone a question they cannot answer?

...would be something like: Ask the entire audience "if I asked you N questions, then for the (N+1)th question asked you 'Is the money in the left door?', would you answer yes?"Truth tellers and liars will both give you the truth, random answerers will not be able to answer.

As stated in the OP any audience member who is not a truth-teller or a liar will randomly lie or tell the truth. If you like, s/he could make that decision by coin flip.

1) Does each member of the audience know the type of all remaining audience member?

2) For the audience members that randomly tell the truth or lie, at any moment, do they know whether they will lie or tell the truth to the next question? That is, before hearing a particular question, have they already decided on lying or telling the truth?

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1) Does each member of the audience know the type of all remaining audience member?

2) For the audience members that randomly tell the truth or lie, at any moment, do they know whether they will lie or tell the truth to the next question? That is, before hearing a particular question, have they already decided on lying or telling the truth?

1. I'm thinking that they do not know much if anything about each other.

They all got complimentary tickets and just arrived for the show.

without an evening together to plot a strategy.

2. I'm not sure, but probably not. But hey, do any of us? .

Specifically, have they already decided? Not necessarily.

OK. Let's say they have not decided anything prior to hearing the question.

An adequate description of the so-called "random" folk is to say that they randomly tell the truth or lie. That choice could be made [a] by coin flip, at any time prior to giving their response, and [c] with or without consultation.

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araver has it. Nice solve. No lie.

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the audience is a red herring. Assuming a single audience member who can be truthteller, liar or undecided/human

, a single question that uses the truth value of the answer inside the question might do the trick for going away with the whole prize all the time. Higher order logic though, bu then so are the axioms about the three types of people.

Truth teller says yes/no equivalent to prize being behind door 1/ being before door 2

Liar would give opposite answer to what he would normally evaluate the expression. If a and b then c.

If prize is behind door one and he would lie his answer would be no. So trying to lie about the outcome pf the hypothetical question, he says yes. Similar argument for the other case.

Undecided person is forced by the paradox of the question to evaluate the trith value of the actual answer before evaluating the answer itself. A machine might get syntactically stuck but a person shoul be able to evaluate/decide the truth value / context before evaluating the question itself. Therefore after he/she does that, the evaluation goes a deterministic path. Hence his/her answer would match the same equivalence as the other types discussed above.

Key here is if this is allowed by the OP.

The wording I intepreted would suggest this apparent paradox would qualify.

Then again, I've already decided if I'm gonna lie or tell the truth in this post

Effectively, by stating your question in this fashion you are forcing the random answerer to either tell the truth or lie twice in a row and thus making his answer deterministic, but then that violates his indeterministic nature of randomness.

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Effectively, by stating your question in this fashion you are forcing the random answerer to either tell the truth or lie twice in a row and thus making his answer deterministic, but then that violates his indeterministic nature of randomness.

The responses of T and L are the same, and R acts like one or the other.

It was wrongfully stated by some, in the discussion, that R randomly chooses Yes and No.

To me, the most elegant solution is to ask the question to the entire audience.

It takes only a moment to deduce that a unanimous answer is reliable.

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I see k-man's point: if the randomizer is really indifferent to whether his/her answer is true or false, then why would he/she be "forced by the paradox to evaluate the truth value..." Why not just flip the coin? There's no requirement that the randomizer actually knows the truth or falsity of his/her answer.

I'm reminded of an old Superboy episode (yes, way old...) where a special kind of Kryptonite made Superboy into Pinocchio--whenever he lied, his nose grew longer. Superboy used this to figure out where the bad guys were hiding, by uttering the words "the bad guys are North of Main Street". When his nose grew longer, he knew that the bad guys were actually South of Main Street. This was an odd theory of "lying"--saying something that doesn't happen to be true, even though you don't know it. But our randomizer could be this way.

(Actually, I generally interpret one of these "randomizers" as being more like Maxwell's daemon--he does something pathological, whatever will best thwart your intended theory)

Now Bonanova's answer has come in. OK, I can bear that interpretation--the randomizer makes an explicit choice between lying and telling the truth. Having made that choice, he/she figures out what IS the truth, and then tells the truth or lies. Humph.

Yes, having a unanimous crowd is very pleasing!

Edited by CaptainEd
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Some solvers were discussing questions to which R could not respond.

What door would you choose in that case?

araver's approach is a solution, in that R can not respond erroneously.

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Let me offer the following perspective:

Consider the question “Are you telling the truth?”
If you answer “No” it means that you are lying, from which it follows that you have told the truth. “No” makes the proposition indeterminable in terms of true/false.
On the other hand, the answer “Yes” is consistent with the truth of the statement (if you answer truthfully, then “Yes” is true.) “Yes” is also consistent with the false of the statement (if you answer falsely, then “Yes, I tell the truth” is in fact false.)
Both Truth Teller and Liar cannot answer “No” to the above question, not because they would be acting opposite to their nature, but because that would make their statement undecidable in terms of true/false. Whereas conditions prohibit them from making undeterminable statements. Consequently, while answering with the only possible determinable choice “Yes”, Random may believe he speaks the truth, or lies and his belief would make it so.
In the same way, the only possible determinable answer to the question, “Are you kidding me?” is “No”.

The inner part of the question proposed by Araver states that the prize is behind the door #1, the outer part (evaluated second) asks to equate own statement to the truthfulness of the first evaluation. In this way Araver may rightfully collect the “Best Answer” prize hidden behind one of the doors.

To remove doubts about whether a person would actually make such a statement about the door #1, I would rephrase the question as follows:

Is your answer as honest as the statement: “the prize is behind the door #1”?

Then what you are asking is in effect: “Your statement = X?”, where X can evaluate to True making the only possible answer “Yes”, or False forcing “No” out of everyone.

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Let me offer the following perspective:

Consider the question “Are you telling the truth?”

If you answer “No” it means that you are lying, from which it follows that you have told the truth. “No” makes the proposition indeterminable in terms of true/false.

On the other hand, the answer “Yes” is consistent with the truth of the statement (if you answer truthfully, then “Yes” is true.) “Yes” is also consistent with the false of the statement (if you answer falsely, then “Yes, I tell the truth” is in fact false.)

Both Truth Teller and Liar cannot answer “No” to the above question, not because they would be acting opposite to their nature, but because that would make their statement undecidable in terms of true/false. Whereas conditions prohibit them from making undeterminable statements. Consequently, while answering with the only possible determinable choice “Yes”, Random may believe he speaks the truth, or lies and his belief would make it so.

In the same way, the only possible determinable answer to the question, “Are you kidding me?” is “No”.

The inner part of the question proposed by Araver states that the prize is behind the door #1, the outer part (evaluated second) asks to equate own statement to the truthfulness of the first evaluation. In this way Araver may rightfully collect the “Best Answer” prize hidden behind one of the doors.

To remove doubts about whether a person would actually make such a statement about the door #1, I would rephrase the question as follows:

Is your answer as honest as the statement: “the prize is behind the door #1”?

Then what you are asking is in effect: “Your statement = X?”, where X can evaluate to True making the only possible answer “Yes”, or False forcing “No” out of everyone.

Thanks, Prime. That's an interesting perspective...So, I've been mulling this over in my head trying to convince myself that everything is correct, so here is a slightly different perspective...

The question "Are you telling the truth?" can be interpreted in different ways. There could be a context (like an answer to a preceding question) or this question could imply "Do you always tell the truth?". In both these cases the Random can answer "No" without creating a paradox. A liar will still answer "Yes" to this question in any context. In a pure case, without any prior context and without "always" being implied, this question indeed forces everybody to answer "Yes", but it's not a useful question anymore. It becomes a rhetorical question and you already know what the answer will be regardless of who is answering. You cannot gain any knowledge from it.

Now, to make it useful, let's add some context to it:

This forces everybody to perform 2 true/false evaluations. Truthteller and Liar are consistent and deterministic. As we've seen many times in the past, it's not hard to force the Liar to tell the truth.

Truthteller: T+T=T.

Liar: L+L=T.

But, to force a Random into a deterministic path is not something I've seen before. A Random can produce 4 combinations of truths and lies leading to 2 different outcomes:

T+T=T, T+L=L, L+T=L and L+L=T

So, now we eliminate some of these combinations by requiring that the truthfullness of both was the same. This eliminates T+L=L and L+T=L as possible answers to the question, just like "No" is an impossible answer to the rhetorical question "Are you telling the truth?". This leaves the Random with only two valid choices that both lead to the truthful answer. Unlike the truthteller and the liar he still has a choice. He can get there by lying twice or telling the truth twice, but other choices have been eliminated.

Very nice puzzle and a great solve, Araver!

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Let me offer the following perspective:

Consider the question “Are you telling the truth?”

If you answer “No” it means that you are lying, from which it follows that you have told the truth. “No” makes the proposition indeterminable in terms of true/false.

On the other hand, the answer “Yes” is consistent with the truth of the statement (if you answer truthfully, then “Yes” is true.) “Yes” is also consistent with the false of the statement (if you answer falsely, then “Yes, I tell the truth” is in fact false.)

Both Truth Teller and Liar cannot answer “No” to the above question, not because they would be acting opposite to their nature, but because that would make their statement undecidable in terms of true/false. Whereas conditions prohibit them from making undeterminable statements. Consequently, while answering with the only possible determinable choice “Yes”, Random may believe he speaks the truth, or lies and his belief would make it so.

In the same way, the only possible determinable answer to the question, “Are you kidding me?” is “No”.

The inner part of the question proposed by Araver states that the prize is behind the door #1, the outer part (evaluated second) asks to equate own statement to the truthfulness of the first evaluation. In this way Araver may rightfully collect the “Best Answer” prize hidden behind one of the doors.

To remove doubts about whether a person would actually make such a statement about the door #1, I would rephrase the question as follows:

Is your answer as honest as the statement: “the prize is behind the door #1”?

Then what you are asking is in effect: “Your statement = X?”, where X can evaluate to True making the only possible answer “Yes”, or False forcing “No” out of everyone.

Thanks, Prime. That's an interesting perspective...So, I've been mulling this over in my head trying to convince myself that everything is correct, so here is a slightly different perspective...

The question "Are you telling the truth?" can be interpreted in different ways. There could be a context (like an answer to a preceding question) or this question could imply "Do you always tell the truth?". In both these cases the Random can answer "No" without creating a paradox. A liar will still answer "Yes" to this question in any context. In a pure case, without any prior context and without "always" being implied, this question indeed forces everybody to answer "Yes", but it's not a useful question anymore. It becomes a rhetorical question and you already know what the answer will be regardless of who is answering. You cannot gain any knowledge from it.

Now, to make it useful, let's add some context to it:

This forces everybody to perform 2 true/false evaluations. Truthteller and Liar are consistent and deterministic. As we've seen many times in the past, it's not hard to force the Liar to tell the truth.

Truthteller: T+T=T.

Liar: L+L=T.

But, to force a Random into a deterministic path is not something I've seen before. A Random can produce 4 combinations of truths and lies leading to 2 different outcomes:

T+T=T, T+L=L, L+T=L and L+L=T

So, now we eliminate some of these combinations by requiring that the truthfullness of both was the same. This eliminates T+L=L and L+T=L as possible answers to the question, just like "No" is an impossible answer to the rhetorical question "Are you telling the truth?". This leaves the Random with only two valid choices that both lead to the truthful answer. Unlike the truthteller and the liar he still has a choice. He can get there by lying twice or telling the truth twice, but other choices have been eliminated.

Very nice puzzle and a great solve, Araver!

I see there is some ambiguity in my example - not deterministic enough.

Let me try again:

I meant my example as a self-referential question/answer. Classical liar paradox is expressed elegantly as follows:

"This sentence is false."

Where one cannot decide the truthfulness of that sentece.

For this problem:

Where the answer "No" will make the proposition (answer) undecidable. Truth-teller, Liar, and Random alike must answer "Yes" to that question.

By itself it does not yield any information about the identity of the replier. However, it forces a consistent answer. When you add a proper variable to it, an answer to the question will produce the desired information.

"Is your answer to this question = X?" Where X may be True or False.

When the answer is "Yes", we conclude X = True; when the answer is "No" -- X = False.

There was nothing in the statement of the problem requiring Random to be consistent by being dishonest exactly twice, or being honest all the time. He can be truthful or not as he pleases. However, while Random has a bit more freedom than Liar or Truth-teller, he is still limited to the one or the other. Random is not allowed to produce undecidable propositions.

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