[bonanova]

In the Land of Knights and Knaves [LKK], Knights told the truth,

and Knaves always lied. When the rare averitas flu virus infected

some of the inhabitants, their behavior was reversed: sick Knights

began to lie, and infected Knaves had to tell the truth!

Your job is to find out the ones who have been infected so they can

be given the costly AFV vaccine. You've just dealt with six of them,

and now a group of ten LKK citizens is brought to your

office, and they venture the following information:

Who are the sick ones? and who are the Knights?

01. Alan: Jack is lying, or he's a Knight; but not both.

02. Bervin: Jack is lying, or I am a Knave; but not both.

03. Clarence: Harry might be a Knave, but only if he is well. Harry is well.

04. David: Harry might be a Knave, but only if he is sick.

05. Evan: Alan might be a Knave, but only if he is well.

06. Bervin: Alan might be a Knave, but only if he is sick. Alan is sick.

07. Fred: If Harry is sick, then Jack is lying.

08. Fred: Alan is lying.

09. George: Fred is lying.

10. George: No fewer than 4 of us are lying.

11. Evan: No more than 4 of us are Knights.

12. Alan: I am a Knight.

13. Harry: I am a Knave.

14. Ingolf: None of our statements is a paradox.

15. Jack: If 2+2=5, then New York is a small city.

Edit #1: Numbering the statements, to make discussion easier.

Edit #2: Bervin and Clarence change their stories, slightly. [thx to s54 and wb!]

------------------------

If you like this type of puzzle, here are some others, recently posted to this forum.

Knights and Knaves get the flu [bononova]

Honestants and Swindlecants: The Delegation [octopuppy]

Knights, Commoners & Knaves [roolstar]

Four Knight and Knave problems [bonanova]

knights and liars [2] [bonanova]

knights, knaves and liars [bonanova]

3 Princess Puzzle [O'Beckon]

The Barber of Honestants and Swindlecants [spoxjox]

The liar, the truth teller....and the random answerer [Martini]

[Guest]

Oh my goodness... the horror. I'll give it a try.

[Guest]

I think this needs clarification. Here's why:

Any If/then conditional must be defined. The IF has to be true for the THEN to be defined. When the IF is false and there is no definition for a false IF, then you have a null state. If you eliminate the null state from what the Knights and Knaves are saying, then you have "False IF = not false" because for the THEN to be false it must be defined. Therefore a False IF = not false = true in the terms of knights and knaves.

That given, Jack has given us a false IF. Therefore his statement MUST be not false becuase the conditional for false is undefined. So Jack = T for our purposes.

Then Fred says if Harry is sick, Jack is lying. This cannot be a false Harry is sick, because that again is an undefined conditional making this true. Therefore this must be true. If Harry is sick, Jack is lying. But Jack is not lying, we know that, so Harry cannot be sick. But Harry says I am a knave which is something a well Knight or Knave cannot say.

The conditionals need to have a false definer. "IF X is true then Y, else Z." Otherwise, you have a logical fallicy becuase when X is true then Y, and X is false, then NOT Y does not logically follow, unless defined that way from the beginning.

Think of it this way - If it's raining, there's water on the ground. If it's true that it's raining, then it's true that there is water in the ground. But, if it's false that it's raining, then you cannot say, therefore, that there is not water on the ground. I might have used a hose. You have to define the False of IF.

See what I mean?

[Guest]

Maybe I'm doing something wrong... but I couldn't find a unique solution.

A = Healthy Knight

B = Sick Knave

C = Healthy Knight or Sick Knave

D = Healthy Knave or Sick Knight

E = Healthy Knave or Sick Knight

F = Healthy Knave or Sick Knight

G = Healthy Knight or Sick Knave

H = Sick Knight

I = Healthy Knight or Sick Knave

J = Healthy Knight

As long as 2 or more of (C,D,E,F,G,I) are Knights, all seem like valid solutions. So I can't definitively answer "Who are the sick ones? and who are the Knights?"

Also, I'm thinking "None of our statements is a paradox" is a true statement, and that an "if X then Y" statement is true unless X is true and Y is false, as per http://regentsprep.org/Regents/Math/tables/ifthen.htm.

Care to give me a clue? Thanks for a good puzzle bonanova!

[Guest]

I think this needs clarification. Here's why:

Any If/then conditional must be defined. The IF has to be true for the THEN to be defined. When the IF is false and there is no definition for a false IF, then you have a null state. If you eliminate the null state from what the Knights and Knaves are saying, then you have "False IF = not false" because for the THEN to be false it must be defined. Therefore a False IF = not false = true in the terms of knights and knaves.

That given, Jack has given us a false IF. Therefore his statement MUST be not false becuase the conditional for false is undefined. So Jack = T for our purposes.

Then Fred says if Harry is sick, Jack is lying. This cannot be a false Harry is sick, because that again is an undefined conditional making this true. Therefore Harry must be sick, but Jack cannot be lying.

The conditionals need to have a false definer. "IF X is true then Y, else Z." Otherwise, you have a logical fallicy becuase when X is true then Y, and X is false, then NOT Y does not logically follow, unless defined that way from the beginning.

Think of it this way - If it's raining, there's water on the ground. If it's true that it's raining, then it's true that there is water in the ground. But, if it's false that it's raining, then you cannot say, therefore, that there is not water on the ground. I might have used a hose. You have to define the False of IF.

See what I mean?

According to

http://regentsprep.org/Regents/Math/tables/ifthen.htm, "IF...THEN is only FALSE when T implies F. All other cases are TRUE." Which is what I also seem to remember from when I took a logic class in college.

From that, you could say that any person who is lying cannot make any sort of "if (false)" statement. They can only make "if (true) then false" statements.

[Guest]

Ah wait.... I think I see a flaw in my logic with Harry's statement. More to come soon...

(posted just before I read the above, which is the flaw I saw. Let me give it another shot.)

[bonanova]

If A then B is logically equivalent to the statement ^A or B.

That is, False is the universal implicator.

True is the universal implicand.

[bonanova]

Maybe I'm doing something wrong... but I couldn't find a unique solution.

A = Healthy Knight

B = Sick Knave

C = Healthy Knight or Sick Knave

D = Healthy Knave or Sick Knight

E = Healthy Knave or Sick Knight

F = Healthy Knave or Sick Knight

G = Healthy Knight or Sick Knave

H = Sick Knight

I = Healthy Knight or Sick Knave

J = Healthy Knight

As long as 2 or more of (C,D,E,F,G,I) are Knights, all seem like valid solutions. So I can't definitively answer "Who are the sick ones? and who are the Knights?"

Also, I'm thinking "None of our statements is a paradox" is a true statement, and that an "if X then Y" statement is true unless X is true and Y is false, as per http://regentsprep.org/Regents/Math/tables/ifthen.htm.

Care to give me a clue? Thanks for a good puzzle bonanova!

Off to the gym; will check back later.

My solution agrees with your statements about A G H I and J, and disagrees with your statements about B - F.

I disagree with your statement that begins, As long as 2 or more of .... are knights...

I agree with your statement about the paradox statement.

I agree with your truth values for If - Then statements.

I think the solution is unique.

I can post it to you in a PM if you like, or give you some more time.

I'm not ruling out the possibility that you are right and I made a mistake,

but I've read my solution a couple of times and it seems right.

Number the statements of the puzzle 1-15 and consider them in this order:

12. 13. 7. 8. 15. 3. 4. 5. 6. 14. 10. 9. 1. 2. 11.

Each statement should give you a piece of info that sets up the following statements.

[Guest]

Couldn't get it either. This is as close as I came:

Alan: Jack is lying, or he's a Knight; but not both.

Jack is telling truth, so he’s a Well Knight

Bervin: Jack is lying, or I am a Knave; but not both.

Bervin tells the truth, and Jack is telling truth, so Bervin is a Sick Knave.

Clarence: Harry might be a Knave, but only if he is well.

(Can’t be a Well Knave) this is false.

David: Harry might be a Knave, but only if he is sick.

(this is true, harry can only be a knave if sick)

Evan: Alan might be a Knave, but only if he is well.

Alan can only be a Knave if sick, So Evan is lying.

Bervin: Alan might be a Knave, but only if he is sick.

Alan can only be a Knave if sick, so Bervin is telling truth.

Fred: If Harry is sick, then Jack is lying.

T=Harry well. Harry can’t be well so this is a T-F statement that is a lie. Fred is lying.

Fred: Alan is lying.

Alan is telling the truth. Fred Lies

George: Fred is lying.

(Fred is lying this is true)

George: No fewer than 4 of us are lying.

(Must be true 4 or more lying)

Evan: No more than 4 of us are Knights.

(Lie – so 4 or more are knights)

Alan: I am a Knight. Well knight.

Harry: I am a Knave.

(Harry is a sick Knight)

Ingolf: None of our statements is a paradox.

Jack: If 2+2=5, then New York is a small city.

If/then conditional based upon false conditional = not false because there is no definition for a false conditional. Assuming no “null” state, this must be true. Well Knight.

(K for Knight N for Knave, W for well S for Sick)

A- WK

B- SN

C- SK or WN

D- WK or SN

E- SK or WN

F- SK or WN

G- WK or SK

H- SK

I- WK or SN

J- WK

[bonanova]

Couldn't get it either. This is as close as I came:

Alan: Jack is lying, or he's a Knight; but not both.

Jack is telling truth, so he’s a Well Knight

Bervin: Jack is lying, or I am a Knave; but not both.

Bervin tells the truth, and Jack is telling truth, so Bervin is a Sick Knave.

Clarence: Harry might be a Knave, but only if he is well.

(Can’t be a Well Knave) this is false.

David: Harry might be a Knave, but only if he is sick.

(this is true, harry can only be a knave if sick)

Evan: Alan might be a Knave, but only if he is well.

Alan can only be a Knave if sick, So Evan is lying.

Bervin: Alan might be a Knave, but only if he is sick.

Alan can only be a Knave if sick, so Bervin is telling truth.

Fred: If Harry is sick, then Jack is lying.

T=Harry well. Harry can’t be well so this is a T-F statement that is a lie. Fred is lying.

Fred: Alan is lying.

Alan is telling the truth. Fred Lies

George: Fred is lying.

(Fred is lying this is true)

George: No fewer than 4 of us are lying.

(Must be true 4 or more lying)

Evan: No more than 4 of us are Knights.

(Lie – so 4 or more are knights)

Alan: I am a Knight. Well knight.

Harry: I am a Knave.

(Harry is a sick Knight)

Ingolf: None of our statements is a paradox.

Jack: If 2+2=5, then New York is a small city.

If/then conditional based upon false conditional = not false because there is no definition for a false conditional. Assuming no “null” state, this must be true. Well Knight.

(K for Knight N for Knave, W for well S for Sick)

A- WK

B- SN

C- SK or WN

D- WK or SN

E- SK or WN

F- SK or WN

G- WK or SK

H- SK

I- WK or SN

J- WK

Alan can only be a Knave if sick, So Evan is lying.

Could a sick Knave claim to be a Knight?

I think what you say is true of Harry, but not of Alan.

[Guest]

I can post it to you in a PM if you like, or give you some more time.

I'll take more time, thanks. I'll give it another shot tomorrow.

[Guest]

Ok I was definitely handling "only if" statements incorrectly. I was treating them as regular ifs. So my solution is definitely incorrect. However, your solution may be incorrect as well. You said:

My solution agrees with your statements about A G H I and J, and disagrees with your statements about B - F.

A subset of that:

A = Healthy Knight

B = Sick Knave

J = Healthy Knight

You agree with the statements about A and J but not B. So let's assume A and J are true for now.

"06. Bervin: Alan might be a Knave, but only if he is sick." This is a true statement. So we know that:

B = Healthy Knight or Sick Knave

But then "02. Bervin: Jack is lying, or I am a Knave; but not both." eliminates the possibility that B is a Healthy Knight. So B must be a Sick Knave.

So if A and J are Healthy Knights, B is a Sick Knave. Let me know your thoughts.

[bonanova]

Ok I was definitely handling "only if" statements incorrectly. I was treating them as regular ifs. So my solution is definitely incorrect. However, your solution may be incorrect as well. You said:

A subset of that:

A = Healthy Knight

B = Sick Knave

J = Healthy Knight

You agree with the statements about A and J but not B. So let's assume A and J are true for now.

"06. Bervin: Alan might be a Knave, but only if he is sick." This is a true statement. So we know that:

B = Healthy Knight or Sick Knave

But then "02. Bervin: Jack is lying, or I am a Knave; but not both." eliminates the possibility that B is a Healthy Knight. So B must be a Sick Knave.

So if A and J are Healthy Knights, B is a Sick Knave. Let me know your thoughts.

05. Clarence: Harry is well.

07. Bervin: Alan is sick.

It seems I made Bervin's statement logically ambiguous.

Alan says I am a Knight. This means Alan is either a HK or a HN.

Based solely on Alan's statement, Alan could be a Knave, but only if he's healthy.

What I want Bervin to do is deny that, and identify himself as a liar.

So I have him say: Alan might be a Knave, but only if he is sick.

But logically Bervin's statement is false only if its premise is true: that Alan could be a Knave.

But later Jake and Fred identify Alan as a Knight, and that makes Bervin's premise false.

Now Bervin's statement is FALSE only if FALSE, and Bervin is no longer lying.

So a false statement becomes a true statement. Ugh.

I need to re-craft statement 07 by Bervin and the similar one meant

to make Clarence a liar - statement 05 - about Harry being a Knave.

I'll post the fix after I'm pretty certain it works.

Probably this is the simplest:

05. Clarence: Harry is a healthy Knave. Or better, Harry is healthy.

07. Bervin: Alan is a sick Knave. Or better, Alan is sick.

Meantime, if you want to continue the analysis using those new statements,

which should identify both Bervin and Clarence as liars, I'd be interested to

know if you come up with a single solution.

Later ...

- BN

Please see the OP for this fix:

[Guest]

Self-evident from initial statements:

12 (Alan) WELL

13 (Harry) SICK -> 4 (David) TRUE

14 (Ingolf) TRUE

15 (Jack) TRUE

Therefore

7 (Fred) FALSE

which directly implies

9 (George) TRUE

10 (George) TRUE

8 (Fred) FALSE -> Alan is a WELL KNIGHT [-> 5 (Evan) TRUE]

1 (Alan) TRUE -> Jack is a WELL KNIGHT

6 (Bervin) FALSE

2 (Bervin) FALSE -> Bervin is a SICK KNIGHT

From 10 (true) we know there are 4 liars. A, D, E, G, I and J are known to be true so our liars are B, C, F and H.

That makes Harry a SICK KNIGHT. Evan (11) is TRUE, and our 4 knights are A, B, H and J.

So we have:

Alan: WELL KNIGHT

Bervin: SICK KNIGHT

Clarence: WELL KNAVE

David: SICK KNAVE

Evan: SICK KNAVE

Fred: WELL KNAVE

George: SICK KNAVE

Harry: SICK KNIGHT

Ingolf: SICK KNAVE

Jack: WELL KNIGHT

[bonanova]

Self-evident from initial statements:

12 (Alan) WELL

13 (Harry) SICK -> 4 (David) TRUE

14 (Ingolf) TRUE

15 (Jack) TRUE

Therefore

7 (Fred) FALSE

which directly implies

9 (George) TRUE

10 (George) TRUE

8 (Fred) FALSE -> Alan is a WELL KNIGHT [-> 5 (Evan) TRUE]

1 (Alan) TRUE -> Jack is a WELL KNIGHT

6 (Bervin) FALSE

2 (Bervin) FALSE -> Bervin is a SICK KNIGHT

From 10 (true) we know there are 4 liars. A, D, E, G, I and J are known to be true so our liars are B, C, F and H.

That makes Harry a SICK KNIGHT. Evan (11) is TRUE, and our 4 knights are A, B, H and J.

So we have:

Alan: WELL KNIGHT

Bervin: SICK KNIGHT

Clarence: WELL KNAVE

David: SICK KNAVE

Evan: SICK KNAVE

Fred: WELL KNAVE

George: SICK KNAVE

Harry: SICK KNIGHT

Ingolf: SICK KNAVE

Jack: WELL KNIGHT

You got it.

How was your skiing?

[Guest]

You got it.

How was your skiing?

Lots of fun thanks

So much so, I'm afraid I've been too busy too cook up anything substantial for your delectation. All I can offer for now is this gristly little morsel...

[Guest]

this is confusing