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Recently there had been horses missing from the stables. The owner barely noticed, until his horse was stolen. He has four suspects but was unable to identify the guilty one. They each gave three statements. He knew that several of the statements were false, in fact, he knew that none of the suspects told the same number of true statements as another one. The statements are as follows:

These first three statements are from James:

Statement 1 Henry is the guilty one.

Statement 2 I was on a fishing trip during the last theft.

Statement 3 No one thinks I am guilty.

These second three statements are from Henry.

Statement 1 James is lying, I am not the guilty one.

Statement 2 Fred has been on duty every theft.

Statement 3 Edgar is guilty.

These third three statements are from Fred.

Statement 1 Henry's first statement is true.

Statement 2 I agree with James' first statement.

Statement 3 I am not guilty

These last three statements are from Edgar.

Statement 1 James's alibi is false.

Statement 2 All of Fred's statements are false.

Statement 3 I would not think of committing the crime.

Now, given all these statements, you should find it easy to figure out who the guilty one is.

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    $\begingroup$ none of the suspects told the same number of true statements as another one. that means one of them has 0 true statement, an other one has 1 true statement etc. ? $\endgroup$ – The random guy Aug 4 '15 at 13:37
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    $\begingroup$ Something else, can a non-guilty guy have a false alibi ? $\endgroup$ – The random guy Aug 4 '15 at 13:46
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    $\begingroup$ I always wonder how the investigators in these stories learn how many truths/lies have been told without figuring out who's doing the truthing/lying. $\endgroup$ – Hellion Aug 4 '15 at 15:39
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    $\begingroup$ I guess Wonder Woman helped them. Or more appropriately, helped in trolling them. $\endgroup$ – Kingrames Aug 4 '15 at 21:51
  • $\begingroup$ @CauseCritical Any chance you can shed some light on your thought process for this? $\endgroup$ – Aggie Kidd Aug 5 '15 at 13:58
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Let's call JS1, JS2 and JS3 the 3 James' statements, HS1, HS2 and HS3 the 3 Henry's statements, etc.

If James is guilty

JS1 : false
JS2 : false
JS3 : false (Edgar is accusing him to lie about his alibi on ES1)
HS1 : true
HS2 : true
HS3 : false
FS1 : true
FS2 : false (supposing he knows he is guilty)
FS3 : true
ES1 : true
ES2 : false
ES3 : ??? (he is allowed to think about stealing the horses without really stealing them)
It is impossible because at least one of the suspect always tell the truth.

If Henry is guilty

JS1 : true
JS2 : ???
JS3 : false (Edgar is accusing him to lie about his alibi on ES1)
HS1 : false
HS2 : true
HS3 : false
FS1 : false
FS2 : true
FS3 : true
ES1 : false
ES2 : false
ES3 : ??? (he is allowed to think about stealing the horses without really stealing them)
It is impossible because at least one of the suspect always tell the truth.

If Fred is guilty

JS1 : false
JS2 : ???
JS3 : false (Edgar is accusing him to lie about his alibi on ES1)
HS1 : false
HS2 : false
HS3 : false
FS1 : true
FS2 : false
FS3 : false
ES1 : false
ES2 : false
ES3 : ??? (he is allowed to think about stealing the horses without really stealing them)
It is impossible because at least one of the suspect always tell the truth.

If Edgar is guilty

JS1 : false
JS2 : ???
JS3 : false (Edgar is accusing him to lie about his alibi on ES1)
HS1 : true
HS2 : true
HS3 : true
FS1 : true
FS2 : false
FS3 : true
ES1 : false
ES2 : false
ES3 : false
It is possible if JS2 is true.

Answer : The guilty one is

Edgar

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    $\begingroup$ Sometimes ya gotta truth table it. $\endgroup$ – Going hamateur Aug 4 '15 at 15:04
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    $\begingroup$ That one is also possible if JS2 is false and ES1 is true. So it doesn't matter if James was fishing or not, he just wasn't stealing the horse. Either way the truths and lies work out right. $\endgroup$ – Togashi Aug 4 '15 at 16:15
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    $\begingroup$ "JS3 : false (Edgar is accusing him to lie about his alibi on ES1)" + "It is impossible because at least one of the suspect always tell the truth." nope, accusing him on lying for his alibi does not imply accusing him of actually stealing the horses. $\endgroup$ – o0'. Aug 4 '15 at 18:05
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    $\begingroup$ @Lohoris: Agreed. And it certainly doesn't imply thinking that James did it. $\endgroup$ – ruakh Aug 5 '15 at 2:12
  • $\begingroup$ Maybe you're right, i supposed that Edgar thinks the only reason that James can lie about his alibi was because he was stealing the horses. $\endgroup$ – The random guy Aug 5 '15 at 6:53
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I think that the guilty one is:

Henry, as told by James.

As each suspect tells a different number of true statements (or a different number of lies), we can look at contradictions in each's statements.

For James, it is possible that every statement is a lie, or every statement is true, or any combination. So he could tell 0, 1, 2, or 3 lies.

For Henry, if we assume the second statement is accusing Fred of being guilty as he was on duty every theft, then 2 and 3 contradict each other, meaning at least 1 is a lie. So we can say that Henry tells 1, 2, or 3 lies.

Fred cannot agree with both James' and Henry's first statements, because he would then agree that Henry is guilty and Henry is not guilty. Also, if he is guilty, then Henry could not be guilty, so Fred can only tell 1 or 2 lies.

Since Fred cannot tell 3 lies, at least 1 has to be true, meaning that Edgar's second statement is false. Thus, Edgar can only say 1, 2, or 3 lies.

This leaves us with James as being the only one that can tell all truths. As James' first statement is that Henry is guilty, we have our culprit.

If Henry is guilty, then Fred agrees with James' first statement and is not guilty, so he told 1 lie (2 truths). If we assume that Henry's second statment is an accusation against Fred, then we know that Henry tells 3 lies (0 truths) since he is guilty and Fred and Edgar cannot be. Thus, Edgar needs to tell 2 lies (1 truth). James' alibi must be true and we know that Fred has to tell at least 1 truth, so apparently Edgar shouldn't be trusted either because he thought about stealing horses.

So, prosecute Henry and fire Edgar for good measure.

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Edgar is the guilty one.

Let's first point out that we know there are 4 people and each of them tells a different number of truths. This means someone always tells the truth, someone always lies, and the remaining people tell either 1 lie or 2.

Note next that Fred's 1st and 2nd statements are tautologically false by virtue of the two comments he's addressing being contradictory. This means he's lied at least once.

Similarly, Edgar is lying in his second comment, since if it were the case that Fred was lying about statements 1 AND 2, then we would have another contradiction.

It must then be the case that either Henry or James is completely truthful. Once this is determined, we can eliminate individuals more easily.

We can reduce the final results to:

J: 2 false/1 true

H: 3 true

F: 2 true/1 false

E: 3 false

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Most of the given answers so far make assumptions beyond what we were given. I think the reason for that is that, as far as I can tell, everything works if either Henry or Edgar are guilty, unless you make at least one assumption.

First, the assumptions I won't make:

  • Thinking someone lied about their alibi isn't the same as thinking they're guilty of the crime (and a corollary: his alibi actually can BE false, even if he isn't guilty).
  • Being "on duty" isn't an assumption of guilt. We're not even told where he works! It could very well be an alibi.
  • Just because someone didn't state that they think someone else is guilty, doesn't mean they don't THINK someone else is guilty.

The one assumption that I am willing to make is that none of these people have serious mental illnesses. So, I think it's reasonable to accept that Fred won't know that James is lying about Henry, but still agree with him, or that Edgar couldn't commit the crime without thinking about it.

So, let's start with the easy part. Fred's first 2 statements are opposites, so one is true and one is false. This means that Edgar's 2nd statement is false. Since neither of them can have 3 true statements, either James or Henry must. Since James says outright that Henry is guilty, and Henry says outright that Edgar is guilty, we know that either Henry or Edgar is the guy. This means that Fred's 3rd statement is true.

Now let's see if it works if James is the one to tell 3 true statements:

  • Of course, this means James has 3 true
  • This means Henry's 1st and 3rd statement are false, so he either has 0 or 1 true
  • This means Fred's 1st statement is false and 2nd is true. We already know his 3rd is true, so he has 2 true
  • This means Edgar's 1st statement is false. We already know his 2nd is false, so he has either 0 or 1 true

In this scenario, we have no way of confirming or denying Henry's 2nd statement or Edgar's 3rd statement. If one of them is false and one is true, then everything works.

And, if Henry is the one to tell 3 true statements:

  • This means James's 1st statement is false, so he has 0, 1, or 2 true
  • Of course, this means Henry has 3 true
  • This means Fred's 1st statement is true and 2nd is false. We already know his 3rd is true, so he has 2 true
  • This means Edgar's 3rd statement is false. We already know his 2nd is false, so he was either 0 or 1 true

In this scenario, we have no way of confirming or denying James's 2nd statement or Edgar's 1st statement, but we do know that they're opposites. If James's 2nd is true, then Edgar's 1st is false. This means Edgar has 0 true, which means James must have 1 true, in which case, James's 3rd statement must be false. If James's 2nd is false, then Edgar's 1st is true. This means Edgar has 1 true, which means James must have 0 true, which again means that Jame's 3rd statement must be false. Either way, everything still works.

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Basics:

  • There is a single culprit (as stated in the original question "...the guilty one")
  • Each suspect makes a different number of true statements (0, 1, 2, or 3).
  • I'll use J1, J2, J3, H1, H2, etc. to represent the statements from each suspect.

Key Assumptions:

  • A guilty person cannot think someone else is also guilty
  • Suspicion of guilt is not withheld from the statements made to the owner

As most of the other great answers have done, we'll start with Fred: F1 and F2 cannot both be TRUE, and cannot both be FALSE, so Fred has made at least one true statement (and one false statement). Thus, E2 cannot be true, due to above (F1 and F2 being mutually exclusive).

# True Statements:
1 <= Fred < 3 (F T * or T F *)
Edgar < 3 (* F *)

Therefore, either James or Henry must be the suspect with three truthful statements. Thus, Henry (per James) or Edgar (per Henry) is the thief.

Fred, then, is not the thief, so F3 is TRUE, and Fred has told exactly two truths. Since no two suspects can tell the same number of truthful statements, Edgar has told at most one truth.

Fred == 2 (F T T or T F T)
Edgar <= 1 (* F *)

H1 and H3 are linked: Henry cannot lie about H1 and tell the truth about H3. Conversely, he cannot lie about H3 and tell the truth about H1. Both situations would result in neither James nor Henry being completely truthful, which conflicts with the guidelines. Furthermore, since Fred has already been determined to be the suspect with two truthful statements, Henry must be either completely truthful, or he is telling at most one truth.

James == 3 (Henry is guilty)
Henry <= 1 (F * F)
Fred == 2 (F T T)
Edgar <= 1 (* F *)

--OR--

James <= 1 (F * *)
Henry == 3 (Edgar is guilty)
Fred == 2 (T F T)
Edgar <= 1 (* F *)

If Henry is completely truthful (H1, H2, and H3 are all TRUE, along with F1), then J1 == FALSE: J1 and H1 cannot both be true.

James <= 1 (F * *)
Henry == 3 (T T T)
Fred == 2 (T F T)
Edgar <= 1 (* F *)

E3 == FALSE: Edgar cannot commit the crime without thinking about it (unless we're really reaching into sleep-rustling, split personalities, demonic possession, etc.)

James <= 1 (F * *)
Henry == 3 (T T T)
Fred == 2 (T F T)
Edgar <= 1 (* F F)

J3 == TRUE: The only person who even comes close to accusing James is Edgar (even then, he only claims the alibi is false, which conveys no direct accusation). In any event, if Edgar is guilty (per Henry), Edgar would know James didn't do it, so it is indeed the case that no one thinks James is guilty.

Now, however, we reach a conflict in the Henry-is-correct path: If J2 == TRUE, then James and Fred would both have two truths, which cannot occur, per the guidelines. If J2 == FALSE, then E1 == TRUE. This is also a conflict, since we would then have no suspects that were completely untruthful.

Therefore, James is the suspect telling the whole truth, and Henry is the thief:

J1 == TRUE: Henry is guilty
J2 == TRUE: Alibi is true
J3 == TRUE: No one thinks James did it

H1 == FALSE: James is telling the truth
H2 == TRUE/FALSE (opposite of E3): Fred being on duty (or not) is irrelevant
H3 == FALSE: Edgar did not do it

F1 == FALSE: Henry is wrong
F2 == TRUE: James is right
F3 == TRUE: Fred did not do it

E1 == FALSE: The alibi of James holds
E2 == FALSE: Fred made two TRUE statements
E3 == FALSE/TRUE (opposite of H2): Irrelevant if Edgar isn't guilty

(The final outcome of H2 and E3 do not affect guilt of Henry, and there do not appear to be any statements clearly pointing to one being truthful over the other, outside of speculation of which person might be more likely to lie about those scenarios.)

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Let's give each sentence a truth value:

James:

$a$ Henry is the guilty one.

$b$ I was on a fishing trip during the last theft.

$c$ No one thinks I am guilty.

Henry:

$a'$ James is lying, I am not the guilty one.

$d$ Fred has been on duty every theft.

$e$ Edgar is guilty.

Fred - either the first or the second is true, the other is false:

$a'$ Henry's first statement is true.

$a$ I agree with James' first statement.

$h$ I am not guilty

Edgar:

$b'$ James's alibi is false.

0 All of Fred's statements are false. (that's a lie)

$f$ I would not think of committing the crime.


$a, b, c, a', d, e, a', a, h, b', 0, f$

$a + a + a' + a' + b + b' + c + d + e + f + h$ = 6

2 + 1 + $c + d + e + f + h$ = 6 (three of $c, d, e, f, h$ are true)

Either Henry $(a', d, e)$ or James $(a, b, c)$ is telling nothing but the truth, so either Henry or Edgar is guilty, making $h$ true and $a'=e$.

$c + d + a' + f$ = 2
$c + d + a' + b$ = 2v3
$b$ >= $f$
$b'$ + $f$ < 2

if $f$ = 1, $b$ = 1, $a$ = 1, $c$ = 1, $d$ = 0, which is an acceptable result (Henry being guilty).

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The thief:

Edgar

Solution

I went with a VERY basic approach to this problem. I only used the final statement from Edgar: 'I would not think of committing this crime'. Since in order to prove a solution to this problem we have to know whether it is true of false. If Edgar is innocent we have no way to prove the truthfulness of that statement. But, if he is guilty then know it is false. Hence Edgar is the thief.

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