Clever Jack - A logical fairy tale

Question

As so often happens in fairy tales, there once was a king who had an intractable problem, one that even his wisest advisers could not solve. After many years of frustration, the exasperated king eventually declared that any who could solve his dilemma would earn his favor and join his royal court.

And as so often happens in these tales, these decrees eventually reached the ears of an clever commoner named Jack. Jack and his two brothers, Kevin (strong as an ox and nearly as bright) and Lewis (simple-minded but generous) set out, and against all odds found a solution to the kings dilemma.

Unfortunately, as happens too often in these tales, the king had second thoughts about his promises after his crisis was resolved. As mere commoners, Jack and his brothers would ill-suit the Kings royal court, thus he schemed to cheat the brothers out of their reward. He claimed that what he actually said was that the people who solved his dilemma would come to live at the castle, but not necessarily as nobility...

In this vein, he has gathered nine people from around the castle, nobility, but also commoners and servants; and the brothers can choose three of the presented people and take their places at the castle.

The king allows one question to be asked to each person present, but aware of Jacks sharp wit, prevents him from asking any of the questions.

Kevin and Lewis query the nine people, but being much less bright than their brother they fail to ask the most useful questions; and receive the less-than-helpful statements below:

• Andrew: “I am a higher rank than Edward”
• Bernard: “Ivan is a noble”
• Charlie: “Andrew is a lower rank than me”
• David: “Charlie is not the same rank as me”
• Edward: “I am not the same rank as Bernard���
• Frederick: “George is not a servant”
• George: “That's right, I'm not a servant”
• Harold: “Frederick told me he is a servant”
• Ivan: “Harold is a commoner”

Despite this, clever Jack believes he can still best the king, and elevate himself and his brothers to nobility. He quickly deduces the following facts about the people gathered before him:

1. Despite his disinclination to actually grant his promised reward, the king wants to at least appear fair.
   Thus there must be at least three nobles amongst the people presented.

2. Nobles would fear to lose their own position to a commoner, but would find amusement in the idea of their peers losing theirs.
   As such they would lie about themselves, but tell the truth about anyone else in the castle, including other nobles.

3. Commoners would have nothing to gain or lose, but would be jealous of their countrymen being elevated.
   Thus they will lie about nobles to prevent Jack picking them, but otherwise tell the truth about themselves and others.

4. Servants would grasp this opportunity to escape their life of servitude, and do anything to be picked.
   Thus they will lie about themselves and anyone of higher rank than themselves to make their position sound better; though they tell the truth about other servants, who present no competition.

Note that these rules are subjective, and not necessarily consistent; It’s hard for liars to keep their stories straight!

Thus:

• A noble asked if he is the same rank as a servant would say yes, as he lies about himself.
• The same noble asked if the servant is the same rank as him would say no, as he tells the truth about servants.

From this, which three people of the nine presented should Jack choose so that he and his brothers become nobility?

---

(Clarifications moved up from the comments)

Ok, sorry does seem to be more uncertainty here than expected; none of it intended to add complications.

1 - To confirm what people have inferred, the ranks at the castle go Noble > Commoner > Servant. There are no intended "ranks within ranks", eg every noble is the same rank as every other noble.

I don't think it matters, but FWIW Jack and his brothers are considered commoners.

2 - All statements are made with the truth values above, even between individuals presented. Thus "X told me he is a Y" is true only if X could tell Jack he is a Y.

The truth value of the second order statement is only intended to be about whether X truly did say Y, modified by whether or not the speaker of the second order statement lies about people of X's rank.

The intention was not to create a lie from whole cloth (X said Y is a lie because X never said Y nor anything like it).

I have spent 15 minutes trying to put my intentions around this clause into clear language, and am still pretty sure I failed. In retrospect this means I should have avoided that sentence type. Live and learn I guess...

3 - My intended syntax and phrasing for determining who the subject of a statement is (in lieu of looking up grammar and syntax etc) is that a sentence is always about the first person mentioned in it, eg: 'I am better than X' is a sentence about oneself, and will be true or false based on whether one lies about themselves; 'X is better than I am' is a sentence about X (even though it refers to ones self), and is true or false depending whether one lies about people of X's rank.

This seems natural to me, but may not be standard (or even technically correct) - and is probably hell for non-native English speakers. Sorry.

Answer 1

There is exactly one set of ranks consistent with what we have been told (at least, if I am interpreting the rules correctly when it comes to H's statement about what F said):

A,B,E are servants; C,G,I are commoners; D,F,H are nobles. So J,K,L should ask to take the places of D,F,H. [EDITED to add: In fact it seems possible for E to be a commoner instead, but this doesn't change what J,K,L should do. See below.]

Here is a proof.

First, G says he isn't a servant. A noble talking about himself will lie, so G is not a noble. Could he be a servant? No, because then F would tell the truth about him (everyone tells the truth about a servant, apart from himself). So G is a commoner.

Now F says that G is not a servant, and is telling the truth about a commoner. A noble or commoner could do this, but not a servant. So F is not a servant.

H says that F says that F is a servant. Who could actually claim to be a servant? Only a noble, as it turns out. (Commoners tell the truth about themselves and servants lie about themselves.) So H is claiming F is a noble. If H is a noble he tells the truth about everyone else, so this works iff F is also a noble; if H is a commoner he lies about nobles and tells the truth about others, neither of which lets him truthfully call someone else a noble; if H is a servant he tells the truth about other servants (so can't call them nobles) but lies about others (so can call them nobles iff they're commoners). So either F and H are both nobles, or F is a commoner and H is a servant.

In particular, H is not a commoner. So I is lying about him. So I is not a noble (who tells the truth about others). Both commoners and servants tell the truth about (other) servants so H is not a servant; so H is a noble and hence F is also a noble.

Now, B falsely calls I a noble. So B isn't a noble (nobles tell the truth about others); nor is B a commoner (commoners tell the truth about non-nobles). So B is a servant. If I were a servant then B would tell the truth about him; so in fact I is a commoner.

E says (about himself) that he is not of the same rank as B. In other words, he is claiming not to be a servant. Nobles lie about themselves so can't do this; commoners tell the truth about themselves so also can't do it. So E is a servant.

A says (about himself) that he is of higher rank than E. This is in fact exactly the same situation as in the previous paragraph, and the conclusion is the same: A is a servant.

Now C again claims to be of higher rank than Andrew, but this is different because now C is talking about Andrew rather than himself. Everyone tells the truth about (other) servants, so indeed C is not a servant.

D says that C is of different rank. This is impossible if D is a servant (it would be true but servants tell the truth only about other servants) or a commoner (if C is a noble it would be true, but commoners lie about nobles; if C is a commoner it would be false, but commoners tell the truth about other commoners; and C is not a servant). Therefore D is a noble.

At this point we have found the three nobles we need, but we might as well add that nobles tell the truth about others, so (since we already know C is not a servant and D, a noble, says he isn't a noble) C is a commoner. And now we know what everyone is.

I actually found this solution with a hacked-together computer program -- which of course provides some verification that the above is correct. Here is the program, for anyone who wants to check my cross-checking.

for a in range(3):
  for b in range(3):
    for c in range(3):
      # C says (about A) "A<C".
      # If C=2 he tells the truth so A<C.
      # If C=1 and A=2 he lies so A>=C, which works.
      # If C=1 and A<2 he tells the truth so A<C so A=0.
      # If C=0 and A>0 he lies so A<C, impossible.
      # If C=0 and A=0 he tells the truth so 0<0, impossible.
      if c==0 or (c==1 and a==1) or (c==2 and a==2): continue
      for d in range(3):
        # D says (about C): "C!=D".
        # If D=2 he tells the truth so C!=D i.e., C<2.
        # If D=1 and C=2 he lies so C=D, impossible.
        # If D=1 and C<2 he tells the truth so C!=D so C=0 or equivly C<D.
        # If D=0 and C>0 he lies so C=D, contradiction.
        # If D=0 and C=0 he tells the truth so C!=D, contradiction.
        if c>=d: continue
        for e in range(3):
          # A says (about A) "A > E". If A=2 he lies so must have E=2;
          # if A=1 he tells the truth so must have E=0;
          # if A=0 he lies which is OK.
          if a>0 and (a,e) not in ((2,2),(1,0)): continue
          # E says (about E) "B!=E".
          # If E=2 he lies so B=E=2.
          # If E=1 he tells the truth so B!=E.
          # If E=0 he lies so B=E=0.
          if (b,e) not in ((2,2),(1,0),(1,2),(0,0)): continue
          for f in range(3):
            # see commented-out code below: we must have g=1 and then f=1 or f=2
            if f==0: continue
            for g in (1,):
            # for g in range(3):
              # F says (about G): "G>0".
              # If F=2 he tells the truth so G>0.
              # If F=1 he lies if G=2, contradiction, but tells the truth if G<2 so G=1.
              # If F=0 he lies if G>0, contradiction, but tells the truth if G=0, contradiction.
              # if f==0 or (f,g) not in ((2,1),(2,2),(1,1)): continue
              # G says (about G): "G>0".
              # If G=2 he lies, contradiction.
              # If G=1 he tells the truth, OK.
              # If G=0 he lies, OK. BUT F's statement means G=0 is impossible; see above.
              # if g!=1: continue
              for h in range(3):
                # H says (about F) that F says (about F) that F=0.
                # F could say F=0 if F=2 but not if F=1 (commoners tell the truth
                # about themselves) nor of F=0 (servants lie about themselves).
                # So effectively H says that F=2.
                # If H=2 he tells the truth about F so we need F=2.
                # If H=1 and F=2 he lies, so F<2 contradiction.
                # If H=1 and F<2 he tells the truth, so F=2, contradiction.
                # If H=0 and F>0 he lies so F<2 so in fact F=1.
                # If H=0 and F=0 he tells the truth so F=2, contradiction.
                if (f,h) not in ((2,2),(1,0)): continue
                for i in range(3):
                  if (a==2)+(b==2)+(c==2)+(d==2)+(e==2)+(f==2)+(g==2)+(h==2)+(i==2)<3: continue
                  # B says (about I) "I=2". If B=2 he tells the truth so I=2;
                  # if B=1 and I=2 he lies so I<2, contradiction;
                  # if B=1 and I<2 he tells truth so I=2, contradiction;
                  # if B=0 and I>0 he lies so I<2 so I=1;
                  # if B=0 and I=0 he tells truth so I=2, contradiction.
                  if (b,i) not in ((2,2),(0,1)): continue
                  # I says (about H) "H=1".
                  # If I=2 he tells the truth so H=1.
                  # If I=1 he tells the truth if H<2 so H=1; lies if H=2 so H=2.
                  # If I=0 he lies if H>0, requiring H=2; tells truth if H=0, contradiction.
                  if (h,i) not in ((1,2),(1,1),(2,1),(2,0)): continue
                  print (a,b,c,d,e,f,g,h,i)

---

I am EDITING this years later to correct an error spotted by Hemant Agarwal and pointed out in comments below. Fortunately, it doesn't appear to make a difference to the final answer. I do not understand why my (admittedly disgusting) program missed the possibility Hemant points out.

When I say above that Edward must be a servant in order to say "I am not the same rank as Bernard", that is a mistake. Bernard has already been determined to be a servant. Edward can in fact also, as far as this statement goes, be a commoner. Commoners tell the truth about anyone who isn't a noble; Edward is talking about himself, and it is true that he is not the same rank as Bernard. So let's suppose Edward is a commoner and see what follows.

Andrew says "I am a higher rank than Edward". So Andrew can't be a noble (they lie about themselves and what A says is true if A is a noble) or a commoner (they tell the truth about themselves and what A says is false if A is a commoner). But it's OK for Andrew to be a servant (they lie about themselves, and what A says is false is A is a servant). So Andrew is a servant.

Charlie says "Andrew is a lower rank than me". That's consistent with Charlie being a noble (nobles tell the truth about others) or a commoner (commoners tell the truth about others who aren't nobles) but not a servant (servants tell the truth about other servants). We will find out in a moment which of those Charlie actually is.

David says "Charlie is not the same rank as me". If Charlie is a noble then no one of any rank can say this. Nobles tell the truth about other nobles; commoners lie about nobles; servants lie about people of higher rank. So in fact Charlie is a commoner. And now David can't be a commoner (commoners tell the truth about other commoners) or a servant (servants lie about commoners) but can be a noble (nobles tell the truth about other people).

So we have found an alternative configuration: instead of the "ABE servants, CGI commoners, DFH nobles" advertised above, we have AB servants, CEGI commoners, DFH nobles. That is, everything is as before other than E's status as a commoner instead of a servant. And JKL should still ask to take the places of DFH.

Answer 2

I am going to step in late and say that

Gareth McCaughan's answer is almost right...

Let us label people with their initial, subscripted by two letters:

1. Whether they are lying or being truthful as $_L$ or $_T$; and
2. Their rank noble, commoner, or servant as $_N$, $_C$, or $_S$

So, a servant named Xavier making a true statement would be $X_{TS}$.

We have

Gareth's solution as:
$A_{LS}\space B_{LS}\space C_{TC}\space D_{TN}\space E_{LS}\space F_{TN}\space G_{TC}\space H_{TN}\space I_{LC}$

The desired meaning, now confirmed by crcroberts, of $H$ 's statement implies

as Gareth points out, that $H$ 's statement may as well be "$F$ is a Noble"

Now

testing out that labelling works just fine
...however, are there other possible scenarios?

Note that

we could keep everyone the same except make $E_{TC}$ instead
the only affected statements are those of $A$ and $E$:
$A$: “$A$ is a higher rank than $E$”; and
$E$: “$E$ is not the same rank as $B$”

$A_{LS}$ is still a servant lying about himself ($R(A_{LS})\not\gt R(E_{TC})$), and
$E_{TC}$ is now a commoner telling the truth about himself ($R(E_{TC})\neq R(B_{LS})$)
- fitting the rules given.

That is:
$A_{LS}\space B_{LS}\space C_{TC}\space D_{TN}\space E_{TC}\space F_{TN}\space G_{TC}\space H_{TN}\space I_{LC}$

Furthermore

the only statements referring to $A,C,D,$ and $E$ are those made by $A,C,D,$ and $E$:
$A$: “$A$ is a higher rank than $E$”
$C$: “$A$ is a lower rank than $C$”
$D$: “$C$ is not the same rank as $D$”
$E$: “$E$ is not the same rank as $B$”

and the only logical link to the others is what $E$ says about $B$.

So maybe we can keep everyone outside this subgroup (including $B_{LS}$) the same and keep $E_{LS}$ but make $A_{TC}$?

That satisfies the statements made by $A$ and $E$, but $C$ can no longer be a commoner as commoners tell the truth about other commoners yet their statement would be a lie.

If $C$ were a noble then $D$ cannot be a noble since a noble tells the truth about other nobles but the statement made by $D$ would be a lie about another noble.

But $D$ could not be a commoner or a servant either as they do lie about nobles yet the same statement would now be true.

So $C$ must be a servant. Furthermore, due to the statement made by $C$ we have $C_{LS}$ since no one, including $A_{TC}$, is of a lower rank than a servant.

$D$ could now be a truthful noble or a truthful commoner, but Jack thinks the king is appearing to be fair, so $D_{TN}$ to keep $3$ nobles in the whole group.

That is:
$A_{TC}\space B_{LS}\space C_{LS}\space D_{TN}\space E_{LS}\space F_{TN}\space G_{TC}\space H_{TN}\space I_{LC}$

Note that

The two scenarios here have the same $3$ nobles as Gareth's solution, so would not change Jack's decision.

Lastly

Are there any other scenarios (maybe by relaxing the link at $B$?
I could not think of any that worked, so I wrote some code, below, which confirms that these $3$ are the only ones - if it is bug free and my logic is correct, of course. Unlike Gareth's I have split everything out into logical functions, so one may inspect individual scenarios if one so wishes.

from itertools import product

# People
A, B, C, D, E, F, G, H, I = (i for i in range(9))

# Ranks
NOBLE=2
COMMONER=1
SERVANT=0

# Each of the statements as a tuple of relevant information:
# (statementTruthFunction(myRank, otherRank), other, isAboutSelf)
STATEMENTS = ((lambda m, o: m > o           , E    , True ),
              (lambda m, o: o == NOBLE      , I    , False),
              (lambda m, o: o < m           , A    , False),
              (lambda m, o: o != m          , C    , False),
              (lambda m, o: m != o          , B    , True ),
              (lambda m, o: o != SERVANT    , G    , False),
              (lambda m, o: m != SERVANT    , G    , True ),
              (lambda m, o: o == NOBLE      , F    , False),
              (lambda m, o: o == COMMONER   , H    , False))

def wouldTellTruth(myRank, otherRank, isAboutSelf):
    '''Returns True if a statement made by a myRank ranked individual would tell the truth in a
    statement involving another individual of rank otherRank otherwise returns False
    - isAboutSelf should be True if the statements primary subject is the individual making the
      statement and False otherwise.'''
    if isAboutSelf not in (True, False):
        raise ValueError("isAboutSelf must be in (True, False)")
    if otherRank not in (NOBLE, COMMONER, SERVANT):
        raise ValueError("otherRank must be in (NOBLE, COMMONER, SERVANT) = {0}".format({NOBLE, COMMONER, SERVANT}))
    if myRank == NOBLE:
        return not isAboutSelf
    elif myRank == COMMONER:
        return isAboutSelf or otherRank != NOBLE
    elif myRank == SERVANT:
        return not isAboutSelf and otherRank == SERVANT
    else:
        raise ValueError("myRank must be in (NOBLE, COMMONER, SERVANT) = {0}".format({NOBLE, COMMONER, SERVANT}))

def analyseState(ranks, statements=STATEMENTS):
    '''Returns a list of which individuals are telling the truth (True) and which are lying (False)
    if the ranks provided is a possible scenario, otherwise returns None.
    - ranks should be an enumerable of ranks (NOBLE, COMMONER, SERVANT), of the same length, and in
      the same oreder, as statements, representing the rank of each individual.
    - statements should be an enumerable of tuples of:
          statementTruthFunction(myRank, otherRank), other, isAboutSelf
      describing the statements being made by each individual
      where:
          - statementTruthFunction returns the truth of the statement, given the ranks
          - other is the index of the other individual in the statement (in both ranks and statements)
          - isAboutSelf is True if the primary subject of the statement is the indivdual making it.'''
    nNobles = sum(1 for myRank in ranks if myRank == NOBLE)
    if nNobles < 3:
        return
    areTellingTruth = []
    for madeByIndex, (statementTruth, otherIndex, isAboutSelf) in enumerate(statements):
        tt = statementTruth(ranks[madeByIndex], ranks[otherIndex])
        st = wouldTellTruth(ranks[madeByIndex], ranks[otherIndex], isAboutSelf)
        if tt != st:
            return
        areTellingTruth.append(tt)
    return areTellingTruth

def testAllPossibleStates(verbose=True, truthText='LT', classText='SCN', format='{0}{1}'):
    results = []
    for ranks in product((SERVANT, COMMONER, NOBLE), repeat=9):
        areTellingTruth = analyseState(ranks, STATEMENTS)
        if areTellingTruth:
            results.append((ranks, areTellingTruth))
    if verbose:
        for ranks, anlysis in results:
            print(' '.join(format.format(truthText[t], classText[c]) for c, t in zip(ranks, anlysis)))
    else:
        return results

Running in it's default form yields the result (left to right alphabetical):

>>> testAllPossibleStates()
LS LS TC TN LS TN TC TN LC
LS LS TC TN TC TN TC TN LC
TC LS LS TN LS TN TC TN LC
>>>

Answer 3

I ended up with the same list of nobles as Gareth McCaughan's answer, but a different solution.

The nobles are:

David, Frederick, Harold

The full rankings are:

Servants: A, B; Commoners: C, E, G, I; Nobles: D, F, H
or
Servants: B, C, E; Commoners: A, G, I; Nobles: D, F, H

Reasoning:

1. G's Statement

   if G is n, then would lie about self -> !n
   if G is c, then would tell truth -> valid
   if G is s, then would lie about self -> maybe, except
   F wouldn't (if F is n, truth about s,
   if F is c truth about s,
   if F is s, truth about s)
   > G is a commoner
   > F is not a servant

2. B's Statement

   if I is n: B would be telling truth, n (!c, !s)
   if I is c: B lie, (!n, !c,) s
   if I is s: B lie, (!n, !c, !s)
   > I is not a servant > B is not a commoner

3. H's Statement This is the tricky one. When I interpreted the statement as

H: "F is a servant"

I only got two nobles, and couldn't solve it. If, as per Gareth McCaughan's answer, I interpret like:

H: "F: I am a servant"

We end up with...

if F is n: H true, H is n
if F is c: H lie, H is s
if F is s: H lie, invalid
> Either F and H are both nobles, or F is a commoner and H is a servant

4. I's Statement

   if H is n: I lie, c or s
   if H is s: I lie, invalid
   > H is a noble
   > F is a noble (by 3)
   > I is a commoner (by 2)
   > B is a servant (by 2)

5. E's Statement

   if E is n: E truth, invalid
   if E is c: E truth, valid
   if E is s: E lie, valid
   > E is not a noble

6. A's Statement

   if A is n: truth, invalid
   if A is c and E is c: lie, invalid
   if A is c and E is s: truth, valid
   if A is s and E is c: lie, valid
   if A is s and E is s: truth, invalid
   > A is not a noble
   > Either A is commoner and E is servant, or A is servant and E is commoner

7. C's Statement

   By 6, we know A is not a noble. That leaves:
   if C is n and A is c: truth, valid
   if C is n and A is s: truth, valid
   if C is c and A is c: lie, invalid
   if C is c and A is s: truth, valid
   if C is s and A is c: lie, valid
   if C is s and A is s: lie, invalid
   >(C=n, A=c), (C=n, A=s), (C=c, A=s), (C=s, A=c)

8. D's Statement

   if C is n and D is n: lie, invalid
   if C is n and D is c: truth, invalid
   if C is n and D is s: truth, invalid
   if C is c and D is n: truth, valid
   if C is c and D is c: lie, invalid
   if C is c and D is s: truth, invalid
   if C is s and D is n: truth, valid
   if C is s and D is c: truth, valid
   if C is s and D is s: lie, invalid
   > (C=c, D=n), (C=s, D=n), (C=s, D=c)
   > C is not a noble
   > D is not a servant
   > From 7, this leaves: C is commoner, A is servant or C is servant, A is commoner

At this point, we know:

B is servant, F is noble, G is commoner, H is noble, I is commoner

There are at least three nobles, and

By 5, E is not a noble. By 6, A is not a noble. By 8, C is not a noble.
So D must be our final noble, consistent with 8.

But

There are two possible ways for D to be a noble:
C is a commoner, or C is a servant.

If

C is a commoner, then A must be a servant (by 7), then E must be a commoner (by 6)
C is a servant, A must be a commoner (by 7), then E must be a servant (by 6)

However

Either way, our nobles must be D, F, and H.

Answer 4

I believe he should pick:

Bernard, Frederick and Ivan

Looking first at:

• Bernard: “Ivan is a noble”

If Bernard was a commoner, and Ivan was a noble, then:

(because commoners lie about nobles,) Bernard would not say that Ivan a noble.

If Bernard was a commoner, and Ivan was not a noble, then:

(because commoners not not lie about non-nobles,) Bernard would not say that Ivan a noble.

Therefore, since Bernard did say that Ivan was a noble, then

Bernard is not a commoner

If Bernard was a servant, then:

(because servants lie to make themselves sound better,) Bernard would not call Ivan (or anyone) a noble.

Therefore:

Bernard is not a servant.

Therefore:

BERNARD IS A NOBLE

If Ivan was not a noble, then:

(because nobles do not lie about other people,) Bernard would not call Ivan a noble.

If Ivan was a noble, then:

(because nobles do not lie about other people,) Bernard would call Ivan a noble.

So:

IVAN IS A NOBLE

From:

• Ivan: “Harold is a commoner”

Then:

Because Ivan is a noble and does not lie about other people, HAROLD IS A COMMONER.

From:

• Harold: “Frederick told me he is a servant”

Then:

Because Harold is a commoner, he would say that a noble or a servant is a servant but would not lie about a commoner. Therefore Frederick is not a commoner. So Frederick is a noble or a servant.

From:

• Frederick: “George is not a servant”

So:

If Frederick is a noble, George is not a servant (because nobles do not lie about other people).

and

If Frederick is a servant, he woudln't say that someone is not a servant. (Servants woudl happily call anyone a servant to make themselves look better!)

So:

FREDERICK IS A NOBLE

So now, we've already found three nobles, so do we really need to continue...?

Answer 5

Assumption: All nobles are of the same rank. One noble cannot be ranked higher than another noble.

Solution

Look at $B$s statement about $I$. If false, then $I$ is not a noble and $B$ has a reason to lie. Commoners only lie about nobles, nobles only lie about themselves, and servants only lie about nobles and themselves. Since $B$ is talking about $I$ (i.e. not himself) and $I$ is not a noble, $B$ has no reason to lie and must be telling the truth. This leads to a contradiction since we assumed $B$ was lying. Thus, $B$ is telling the truth and $I$ is a noble.

As a consequence, we know that $B$ cannot be a commoner or a servant since he would have lied. Thus, $B$ is also a noble.

$E$ says that he is not the same rank as $B$. If $E$ was a servant, this would be a truth, yet $E$ should have lied according to the rules, so $E$ cannot be a servant. The statement is about both $B$ and $E$, so if $E$ is a commoner, then he should lie about nobles, which he did not do, so $E$ is not a commoner. Thus, $E$ is also a noble.

Since we were only required to find three nobles, we are done! The brothers pick these three to swap with and live on as nobles in the castle!

Caveat

There is a comment that statements of the form "X ... Y" are about X, not Y. Thus, our last deduction is not necessarily correct. $E$'s statement is only about $E$, and thus, $E$ could be a commoner (they don't lie about themselves) or a noble (they do lie about themselves).

$A$ says he is higher rank than $E$. If $A$ spoke true, then $E$ must be a commoner since we already eliminated $E$ as a servant and $E$ as a noble would be a contradiction. But then $A$ should have lied about himself since he was either a servant or a noble talking about himself, both of which lie about themselves. So we know that $A$'s statement is false.

If $A$ is a commoner, then he'd have no reason to lie about himself so he should have told the truth, so $A$ is not a commoner. If $A$ is a servant, then he would lie about himself, and his statement would only be a lie if $E$ was a noble or a servant (but we already know $E$ is not a servant). If $A$ was a noble, then again he would lie about himself. The statement is only a lie if $E$ is also a noble. Thus, $A$ can be either a noble or a servant, but either way $E$ must be a noble.

Thus, even with this rule change, Bernard, Ivan, and Edward are the nobles who the brothers should swap with.

Answer 6

Here is my reasoning:

From the first statements:

• Andrew: “I am a higher rank than Edward

  Andrew=servant -> Edward=any
  Andrew=commoner -> Edward=servant
  Andrew=nobility -> Edward=nobility

• Bernard: “Ivan is a noble”

  Bernard=servant -> Ivan=commoner
  Bernard=commoner -> impossible
  Bernard=nobility -> Ivan=nobility

• Charlie: “Andrew is a lower rank than me”

  Charlie=servant -> Andrew = commoner/nobility
  Charlie=commoner -> Andrew = servant/nobility
  Charlie=nobility -> Andrew =servant/commoner

• David: “Charlie is not the same rank as me”

  David=servant -> impossible
  David=commoner -> Charlie = servant
  David=nobility -> Charlie =servant/commoner

• Edward: “I am not the same rank as Bernard”

  Edward=servant -> Bernard=servant
  Edward=commoner -> Bernard =servant/nobility
  Edward=nobility -> Bernard =nobility

• Frederick: “George is not a servant”

  Frederick=servant -> impossible
  Frederick=commoner -> George = commoner
  Frederick=nobility -> George = commoner/nobility

• George: “That's right, I'm not a servant”

  George = servant ok
  George = commoner ok
  George = nobility impossible

• Harold: “Frederick told me he is a servant”

  1)Frederick: "I am a servant"
  Frederick=servant impossible
  Frederick=commoner impossible
  Frederick=nobilty ok
  2)"he told me that"
  Harold=servant & Frederick=servant -> impossible
  Harold=servant & Frederick=commoner -> ok (lie on the fact that he said it)
  Harold=servant & Frederick=nobilty -> (even if Frederick could have said it, he may have not said anything to Harold)
  Harold=commoner & Frederick=servant -> impossible
  Harold=commoner & Frederick=commoner -> impossible
  Harold=commoner & Frederick=nobilty -> ok (even if Frederick could have said it, he may have not said anything to Harold)
  Harold=nobilty & Frederick=servant -> impossible
  Harold=nobilty & Frederick=commoner ->impossible
  Harold=nobilty & Frederick=nobilty -> ok

• Ivan: “Harold is a commoner”

  Ivan=servant->impossible
  Ivan=commoner-> harold=commoner/nobility
  Ivan=nobilty -> harold =commoner

---

We use each statement:

• Andrew:

  Andrew=servant -> Edward=any
  Andrew=commoner -> Edward=servant
  Andrew=nobility -> Edward=nobility

• Bernard

  Bernard=servant -> Ivan=commoner
  Bernard=nobility -> Ivan=nobility

• Charlie

  Charlie=servant -> Andrew = commoner/nobility
  Charlie=commoner -> Andrew = servant/nobility

• David

  David=commoner -> Charlie = servant
  David=nobility -> Charlie =servant/commoner

• Edward

  Edward=servant -> Bernard=servant
  Edward=nobility -> Bernard =nobility

• Frederick

  Frederick=nobility

• George

  George = commoner

• Harold

  Harold=commoner
  Harold=nobility

• Ivan

  Ivan=commoner-> Harold=commoner/nobility
  Ivan=nobility -> Harold =commoner

---

Now we make a guess: Bernard is/isn't a noble
1) He is

with the information that we have at least 3 nobles we get:
(N = noble , C = commoner , S = servant)
Andrew S/C
Bernard S
Charlie S/C
David N
Edward S
Frederick N
George C
Harold N
Ivan C

2)He isn't

Andrew S/N
Bernard N
Charlie C
David N
Edward N
Frederick N
George C
Harold C
Ivan N

And we got:

We got 2 nobles in common: David and Frederick, but the puzzle states we can find the 3 nobles so I must be wrong somewhere

Answer 7

Jack should name

Bernard,Ivan and Frederick

There are multiple consistent arrangements.

They are: Using the notation A0-Andrew servant, A1-Andrew commoner, A2-Andrew Nobility

D1C0A2E2B2I2H1F2G1
D2C1A2E2B2I2H1F2G1
D2C1A0E2B2I2H1F2G1
D2C1A0E1B2I2H1F2G1

In addtion the following arrangements are consistent with the answers, but not with the requirement that there be 3 nobles.
D1C0A1E0B0I1H1F2G1
D2C0A1E0B0I1H1F2G1
D2C1A0E0B0I1H1F2G1
D2C1A0E1B0I1H1F2G1

However The three named persons are nobles in any valid arrangement.

Going through the first valid sequence, D1C0A2E2B2I2H1F2G1, for consistency:
S1:Lie nobility about self
S2:Truth nobility about other
S3:Lie servant about other non servant
S4:Truth commoner about non nobility
S5:Lie nobility about self
S6:Truth nobility about other
S7:Truth commoner about self
S8:Truth commoner about non nobility

Answer 8

I'm writing up the details but for now I think I have an answer:

He should pick Bernard, David and Ivan

I just scrapped the numbers. I think it made it more confusing.

1 - Let's start with Bernard's statement: if Bernard is noble, the statement is true. If he's a commoner, he can't lie about commoners/servants being noble and also wouldn't confirm a noble being noble. If he's a servant, he wouldn't call anyone other than himself noble as it reduces his chances. This confirms both Ivan and Bernard to be nobles.

2 - Since Ivan is noble his statement is also true: Harold is a commoner.

3 - Harold's statement could mean 2 things: He's telling the truth and Frederick is a servant or he's lying and Frederick is noble(Frederick=Servant/Noble)

4 - Frederick's statement could be stated like "George is a commoner/noble". If he's a servant, he wouldn't call anyone noble, he could however be calling a nobility common. If he's noble, both could be true.(George=Commoner/Noble)

5 - George's Statement: He's saying he's not a servant. If he's noble, he could be saying he's a commoner. If he's a commoner, he could confirm to be one. Anyway, his commoner/noble state doesn't change.

6 - Edward's statement is the same as saying "I am not noble". A servant wouldn't do this. A commoner could say this as it is the truth. A nobility could also say it as they lie about themselves. (Edward=Commoner/Noble)

7 - Andrew's statement can be changed to "I am a higher rank than commoner/noble". A commoner couldn't say this as it isn't true. A nobility wouldn't say this as it confirms him to be noble. This only leaves servant as possibility. (Andrew=Servant)Now for the second part. Since they cannot lie about the 2nd part of the statements or change the facts, it must be actually possible to be a higher rank than Edward. This means Edward is a commoner.

8 - Charlie's statement is about Andrew and not himself. No one lies about servants and since the statement is not about the person himself the whole statement is true. This means Charlie is either a commoner or noble. (Charlie=Commoner/Noble)

9 - Finally we come to David's statement: “Charlie is not the same rank as me”. Again, the statement is about another person meaning David doesn't/can't lie about his own rank. Here I checked all possibilities one by one.
List Form:

 (Servant)         Andrew: “I am a higher rank than Edward”

 (Noble)           Bernard: “Ivan is a noble”

 (Commoner/Noble)  Charlie: “Andrew is a lower rank than me”

 (Noble if Charlie=Commoner, Servant/Commoner if Charlie=Noble)
                   David: “Charlie is not the same rank as me”

 (Commoner)        Edward: “I am not the same rank as Bernard” = "I am not noble"

 (Servant/Noble)   Frederick: “George is not a servant” = "George is a commoner/noble"

 (Commoner/Noble)  George: “That's right, I'm not a servant” "I am a commoner/noble"

 (Commoner)        Harold: “Frederick told me he is a servant”

 (Noble)           Ivan: “Harold is a commoner”

Looking at it multiple times over again, I kinda doubt my first answer was legit. I'm pretty sure about the ones I did find are correct. Just can't figure out how to get the 3rd noble.

Answer 9

I've got the answer .....but before that remember one thing - it is clearly stated in the question that among the nine people there are at least three Nobles among the people (it doesn't mean there are only three,it could be greater than three as well).

Now the answer:

1. Andrew - Noble
2. Bernard - Servant
3. Charlie - Commoner
4. David - Noble
5. Edward - Noble
6. Frederick - Noble
7. George - Commoner
8. Harold - Commoner
9. Ivan - Commoner

Well, accordingly the answer could be any three of

Andrew, David, Edward and Frederick.

I will try to write a short explanation for this later, because explaining this would take ages....

If you are not satisfied with my answer you could crosscheck it.

Answer 10

-->According to me,Its Bernard,David, Fredrick and Ivan.

-->Andrew is a servant trying to make his position sound better.

-->Bernard is a noble telling truth about another noble,since a noble can state truth about anyone in the castle except themselves ,commoners can only state that a noble is a lower rank than them ,can't lie about others and slaves would only try to make their position sound better

-->Charlie is a commoner speaking truth about Andrew

-->David is a noble speaking truth about Charlie

-->Edward is a commoner speaking truth about himself

-->Fredrick is a noble speaking truth about George

-->George is a commoner speaking truth about himself

-->Harold is a commoner telling lies about Fredrick since he can lie about nobles

-->Ivan is already a noble telling truth about Harold