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## Question

In the weird land of KKL, Knights always tell the truth, Knaves alternately tell the truth and lie and Liars always lie.

You have been sent on assignment to KKL to learn the ways of the natives and explore ways of identifying their type.

A guide leads you to a dark alley and introduces you to Al, Bob and Chuck, who agree to answer exactly three questions.

It's too dark to see their faces, you don't know which type of citizen each is, you can't tell which of them is speaking, and they reply in random order. Although they answer anonymously, you must determine which type of citizen they are.

You decide to start with the direct approach and then play it by ear:

[1] I'd like to know what type of citizens I'm talking with. Please tell me.

Sure thing. Myself, I'd be misleading you if I didn't tell you I'm a knight.

Hate to admit it, but I'm the unreliable one, being as how I'm a knave.

Don't listen to anything I tell you mate, I'm a bloody liar.

[2] OK great, thanks for that. But I wonder whether there really is one of each type.

Maybe there's more than one of some type. Can you comment specifically on that?

Pretty shrewed, aren't you? Well, you're right - it turns out we're all the same type.

Forget him. We told you already: there's exactly one of each type here.

Aww, they're both messing with you: actually, there's exactly two of one type here.

[3] OK well that's pretty confusing. For a final question, it might help just to talk about Al.

Would you tell me what type of citizen Al is? I think that will tell me all I need to know.

Why all the attention for Al? He's just a knave.

No way! Al is a liar!

Well I'll tell you this much, whatever Al is, Bert Bob is something else.

OK guys, thanks a lot.

I think now I know what type Chuck is.

And let's see ... what type Al is ... and also Bob.

Are you messing with us, or do you really know?

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I spent a long time typing up an involved reply. A few minutes before finishing it off, my computer reset itself. So I'm not going to attempt a response again until I get home after work, which may be very late in the day or even tomorrow (or later). For now:

There is no unique combination that deductively follows. All could be knaves, for example. There are other combinations that work, too.

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This looks hard

I will attempt soon

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I spent a long time typing up an involved reply. A few minutes before finishing it off, my computer reset itself. So I'm not going to attempt a response again until I get home after work, which may be very late in the day or even tomorrow (or later). For now:

There is no unique combination that deductively follows. All could be knaves, for example. There are other combinations that work, too.

Remember knaves alternately tell the truth and lie.

After their first response they're deterministic.

Hint: use Notepad and save often, then copy into your reply.

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So let me get this right: Knaves do either of the following for three questions:

lie-truth-lie

OR

truth-lie-truth

Am I getting that right?

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Well I'll tell you this much, whatever Al is, Bert is something else.

Do you mean Bob, or is this a lie based on the wrong name, or irrelevant because of the wrong name?

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Great Puzzle.... I think I have it...

Al = Knave

Bob/Bert = Knave

Chuck = Liar

And here is why...

In the weird land of KKL. Knights always tell the truth, Knaves alternately tell the truth and lie and Liars always lie.

You have been sent on assignment to KKL to learn the ways of the natives and explore ways of identifying their type.

A guide leads you to a dark alley and introduces you to Al, Bob and Chuck, who agree to answer exactly three questions.

It's too dark to see their faces, you don't know which type of citizen each is, you can't tell which of them is speaking, and they reply in random order. Although they answer anonymously, you must determine which type of citizen they are.

You decide to start with the direct approach and then play it by ear:

[1] I'd like to know what type of citizens I'm talking with. Please tell me.

Sure thing. Myself, I'd be misleading you if I didn't tell you I'm a knight.

Hate to admit it, but I'm the unreliable one, being as how I'm a knave.

Don't listen to anything I tell you mate, I'm a bloody liar.

Only options are

Respondent #1 - Truth (Knight (???or Knave telling truth???)) or Lie (Knave lying or Liar)

Respondent #2 - Truth (Knave telling truth) or Lie (Liar)

Respondent #3 - Truth (not possible) or Lie (Knave lying) (can't be liar or he'd be tellint truth)

Therefore we can now conclude that respondent #3 = Knave (who is lying)

We can also conclude that this same Knave with respond truthfully to second question.

We can also conclude that only possible types of responders must include at least 1 Knave.

[2] OK great, thanks for that. But I wonder whether there really is one of each type.

Maybe there's more than one of some type. Can you comment specifically on that?

Pretty shrewed, aren't you? Well, you're right - it turns out we're all the same type.

Forget him. We told you already: there's exactly one of each type here.

Aww, they're both messing with you: actually, there's exactly two of one type here.

Only options are

Respondent #1 - Truth (Knave telling truth and then all are Knaves) or Lie (Knave lying or Liar and not all same)

Respondent #2 - Truth (Knight* or Knave telling truth and one of each) or Lie (Knave lying or Liar and not all same and not one of each)

Respondent #3 - Truth (not possible) or Lie (Knave lying) (can't be liar or he'd be tellint truth)

*but question as to if he lied about already telling us forcing this to be a Knave who lied and then told truth in same answer - LET'S IGNORE FOR NOW

---> which actually we can ignore b/c that would require two Knaves telling truth here which is not possible.

---> or if you want to get really critical… means that the 2nd responder must be the liar (he just lied 2x)

From the fact that a Knave lied to the first question… we know we have at least 1 Knave telling truth here.

If a Knight were present thent we'd have two True statements here which is not possible

So we can conclude that only one of these statements is true (and it is a Knave telling the truth).

because all statements are contradictory (1=all same, 2=all different, 3=2 of 1 type)

Therefore we can conclude that we have NO KNIGHTS

and furthermore that in the responses to question #1 the first respondent and third respondent lied!

and furthermore if that respondent were a knave he'd have to tell the truth to the 2nd question

which is not possible b/c we can only have one truth to 2nd question… therefore Respondent #1 to first Question = Liar.

Therefore we know that the only possibilites are Two Knaves and One Liar or One Knave and Two Liars.

Furthermore… if we have additional Knaves… they are lying now… so they must have told truth in response to question #1.

Which means they would tell truth in Response to Question #3 (b/c they would have had to told truth to question #1 as Respondent #2 there)

Or we know that we have 3 lies in response to question #3.

[3] OK well that's pretty confusing. For a final question, it might help just to talk about Al.

Would you tell me what type of citizen Al is? I think that will tell me all I need to know.

Why all the attention for Al? He's just a knave.

No way! Al is a liar!

Well I'll tell you this much, whatever Al is, Bert is something else.

OK guys, thanks a lot.

I think now I know what type Chuck is.

And let's see ... what type Al is ... and also Bob.

Are you messing with us, or do you really know?

From above… only options are 3 Lies (KLL) or 2 Lies (KKL - with one Knave telling truth)

Only options regarding these responses that meet those conditions are

Respondent #1 - Truth (Knave telling truth and Al=Knave) or Lie (Knave lying or Liar and Al=Liar)

Respondent #2 - Truth (Knave telling truth and Al=Liar) or Lie (Knave lying or Liar and Al=Knave)

Respondent #3 - Truth (Knave telling truth and Al<>Bert) or Lie (Knave lying or Liar and Al=Bert)

With regards to Respondents #1 and 2 they contradict and we know Al must be a Knave or a Liar so 1 of them is True.

As such… we must have a Knave telling the truth here… a Knave lying here… and a liar.

So Two Knaves and 1 Liar.

Because #1 or 2 is True this also makes #3 to be a Lie

As such AL=BERT = Knave (I'm assuming that the Bert/Bob issue is a typo)

And Chuck = Liar

Question for you... what is chance Al is the Lying Knave (Lying in response to queries 1 and 3)?

If Resp1=Al then Al = Knave Telling Truth here.

If Resp1<>Al then Al = Knave Lying (2 times out of 2)

If Resp2<>Al then Al = Knave Lying (1 of 2) or Knave Trueing (1 of 2)

If Resp3<>Al then Al = Knave Lying (1 of 2) or Knave Trueing (1 of 2)

2/3 likelihood that Al = the Lying Knave in response to Questions #1 and #3.

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Unreality asked:

So let me get this right: Knaves do either of the following for three questions:

lie-truth-lie

OR

truth-lie-truth

Am I getting that right?

Correct

Writersblock asked:

Do you mean Bob, or is this a lie based on the wrong name, or irrelevant because of the wrong name?

My bad. Bert/Bob is a typo.

Great Puzzle.... I think I have it...

In the weird land of KKL. Knights always tell the truth, Knaves alternately tell the truth and lie and Liars always lie.

You have been sent on assignment to KKL to learn the ways of the natives and explore ways of identifying their type.

A guide leads you to a dark alley and introduces you to Al, Bob and Chuck, who agree to answer exactly three questions.

It's too dark to see their faces, you don't know which type of citizen each is, you can't tell which of them is speaking, and they reply in random order. Although they answer anonymously, you must determine which type of citizen they are.

You decide to start with the direct approach and then play it by ear:

[1] I'd like to know what type of citizens I'm talking with. Please tell me.

Sure thing. Myself, I'd be misleading you if I didn't tell you I'm a knight.

Hate to admit it, but I'm the unreliable one, being as how I'm a knave.

Don't listen to anything I tell you mate, I'm a bloody liar.

Only options are

Respondent #1 - Truth (Knight (???or Knave telling truth???)) or Lie (Knave lying or Liar)

Respondent #2 - Truth (Knave telling truth) or Lie (Liar)

Respondent #3 - Truth (not possible) or Lie (Knave lying) (can't be liar or he'd be tellint truth)

Therefore we can now conclude that respondent #3 = Knave (who is lying)

We can also conclude that this same Knave with respond truthfully to second question.

We can also conclude that only possible types of responders must include at least 1 Knave.

[2] OK great, thanks for that. But I wonder whether there really is one of each type.

Maybe there's more than one of some type. Can you comment specifically on that?

Pretty shrewed, aren't you? Well, you're right - it turns out we're all the same type.

Forget him. We told you already: there's exactly one of each type here.

Aww, they're both messing with you: actually, there's exactly two of one type here.

Only options are

Respondent #1 - Truth (Knave telling truth and then all are Knaves) or Lie (Knave lying or Liar and not all same)

Respondent #2 - Truth (Knight* or Knave telling truth and one of each) or Lie (Knave lying or Liar and not all same and not one of each)

Respondent #3 - Truth (not possible) or Lie (Knave lying) (can't be liar or he'd be tellint truth)

*but question as to if he lied about already telling us forcing this to be a Knave who lied and then told truth in same answer - LET'S IGNORE FOR NOW

---> which actually we can ignore b/c that would require two Knaves telling truth here which is not possible.

---> or if you want to get really critical… means that the 2nd responder must be the liar (he just lied 2x)

From the fact that a Knave lied to the first question… we know we have at least 1 Knave telling truth here.

If a Knight were present thent we'd have two True statements here which is not possible

So we can conclude that only one of these statements is true (and it is a Knave telling the truth).

because all statements are contradictory (1=all same, 2=all different, 3=2 of 1 type)

Therefore we can conclude that we have NO KNIGHTS

and furthermore that in the responses to question #1 the first respondent and third respondent lied!

and furthermore if that respondent were a knave he'd have to tell the truth to the 2nd question

which is not possible b/c we can only have one truth to 2nd question… therefore Respondent #1 to first Question = Liar.

Therefore we know that the only possibilites are Two Knaves and One Liar or One Knave and Two Liars.

Furthermore… if we have additional Knaves… they are lying now… so they must have told truth in response to question #1.

Which means they would tell truth in Response to Question #3 (b/c they would have had to told truth to question #1 as Respondent #2 there)

Or we know that we have 3 lies in response to question #3.

[3] OK well that's pretty confusing. For a final question, it might help just to talk about Al.

Would you tell me what type of citizen Al is? I think that will tell me all I need to know.

Why all the attention for Al? He's just a knave.

No way! Al is a liar!

Well I'll tell you this much, whatever Al is, Bert is something else.

OK guys, thanks a lot.

I think now I know what type Chuck is.

And let's see ... what type Al is ... and also Bob.

Are you messing with us, or do you really know?

From above… only options are 3 Lies (KLL) or 2 Lies (KKL - with one Knave telling truth)

Only options regarding these responses that meet those conditions are

Respondent #1 - Truth (Knave telling truth and Al=Knave) or Lie (Knave lying or Liar and Al=Liar)

Respondent #2 - Truth (Knave telling truth and Al=Liar) or Lie (Knave lying or Liar and Al=Knave)

Respondent #3 - Truth (Knave telling truth and Al<>Bert) or Lie (Knave lying or Liar and Al=Bert)

With regards to Respondents #1 and 2 they contradict and we know Al must be a Knave or a Liar so 1 of them is True.

As such… we must have a Knave telling the truth here… a Knave lying here… and a liar.

So Two Knaves and 1 Liar.

Because #1 or 2 is True this also makes #3 to be a Lie

As such AL=BERT = Knave (I'm assuming that the Bert/Bob issue is a typo)

And Chuck = Liar

You got it. Nice work.

Question for you... what is chance Al is the Lying Knave (Lying in response to queries 1 and 3)?

If Resp1=Al then Al = Knave Telling Truth here.

If Resp1<>Al then Al = Knave Lying (2 times out of 2)

If Resp2<>Al then Al = Knave Lying (1 of 2) or Knave Trueing (1 of 2)

If Resp3<>Al then Al = Knave Lying (1 of 2) or Knave Trueing (1 of 2)

2/3 likelihood that Al = the Lying Knave in response to Questions #1 and #3.

Al = Knave

Bob/Bert = Knave

Chuck = Liar

And here is why...

Interesting question.

I can't distinguish between Al and Bob.

Let:

Knave1 [the TFT knave] = K1;

Knave2 [the FTF knave] = K2;

Liar = C

What are the chances that Al = K2?

Same as the chances that Al = K1:

Answers for Q1: FTF - C / K1 / K2 <==> either C/Al/Bob or C/Bob/Al. Equally likely

Answers for Q2: FFT - C or K1 / K1 or C / K2 <==> C/K1/K2 or K1/C/K2. <==> C/Al/Bob, C/Bob/Al, Al/C/Bob or Bob/C/Al. Equally likely.

Answers for Q3: TFF - K1 / K2 or C / C or K2 <==> K1/K2/C or K1/C/K2. <==> Al/Bob/C, Bob/Al/C, Al/C/Bob or Bob/C/Al. Equally likely.

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