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There is a broken window in the house. I asked who broke the window:

Gregory: I was not!

April : I didn’t do it.

August: April was.

June : August says the truth

Two of them said the truth, and two of them lied.

I asked them again:

Gregory: It was June

April: August didn’t do it

August: April didn’t do it!

June: Gregory was lying.

Who broke the window? Please explain the logic.

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  • $\begingroup$ Welcome to PSE! Take the tour if you haven't already or have a look at the help section if you have any questions. Hope you enjoy your time here! $\endgroup$
    – Christoph
    Aug 10 '20 at 7:36
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    $\begingroup$ This doesn't seem completely solvable with the information given. Is there any more info you haven't added? E.g. was the window broken by exactly one person or more than one, must that person be among the four speakers, what do we know about the truth/lie values of the second set of statements? $\endgroup$ Aug 10 '20 at 7:46
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    $\begingroup$ It'd be good to get a grammar edit by OP just to clarify what they mean with Gregory and August's first statements. The word 'was' doesn't fit the sentence structure - I assume Gregory is saying 'I didn't do it" and August is saying 'April did it', but it'd be good to confirm that. $\endgroup$
    – Kayndarr
    Aug 10 '20 at 7:50
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    $\begingroup$ I think it also needs clarification if a liar can still tell the truth? Otherwise August is a contradiction since of the two statements he made, exactly one must be true. $\endgroup$
    – Cain
    Aug 10 '20 at 16:21
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Gregory: I was not!

April : I didn’t do it.

August: April was.

June : August says the truth

Two of them said the truth, and two of them lied.

If June or August told the truth, then so did the other, so April broke the window and Gregory is lying, so it was broken by April and Gregory, and maybe also an outsider. That's one possibility.

Otherwise, April and Gregory told the truth, because two did. Then it was broken by August and/or June and/or an outsider.

So, in sum,

it was broken by April and Gregory together, or by August and/or June. And maybe an outsider helped, too. Or indeed maybe an outsider did it alone.

The second batch of statements tells us nothing, since we don't know how many people are telling the truth that time.

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Development

I will try to use a mix between the formal notation and natural language for convenience. This is a long development of rational logic. If you want to skip to the judging part, jump to the bottom.

Let:

Gr = Gregory;
Ap = April;
Au = August;
Ju = June;
A = First set of answers;
B = Second set of answers;
~ = Negative (turns the premisse false);
T = True value;
F = False value

Premise 1: There is a broken window in the house.

Question A: Who broke the window?

GrA:  ~Gr         (Gregory was not)
ApA:  ~Ap         (April was not)
AuA:  Ap          (April was)
JuA:  AuA -> Ap   (AuA is true, therefore April was)

We can't conclude anything from this but if we take the major opinion it was April who broke the window and therefore April is lying.

However, we were presented with a new premise:


New premise

Premise 2: Two answers to question A are truth and two are false.

Truth table yelds six possibilites:

A   GrA   ApA   AuA   JuA
--------------------------
1   T     T     F     F
2   T     F     T     F
3   T     F     F     T
4   F     T     T     F
5   F     T     F     T
6   F     F     T     T

We can't have AuA and ~AuA, therefore the cases where AuA and JuA have different truth values are contradictions. Let's try to draw it more formally:

A1: ~Gr + ~Ap + ~(Ap) + ~(AuA) -> ~(Ap) => ~Gr + ~Ap (Gregory and April did not break the window.)
A2: ~Gr + ~(~Ap) + Ap + ~(AuA) -> ~(Ap) => (Contradiction - We can't have AuA and ~AuA.)
A3: ~Gr + ~(~Ap) + ~(Ap) + AuA -> Ap => (Contradiction - We can't have AuA and ~AuA.)
A4: ~(~Gr) + ~Ap + Ap + ~(AuA) -> ~(Ap) => (Contradiction - We can't have AuA and ~AuA.)
A5: ~(~Gr) + ~Ap + ~(Ap) + AuA -> Ap => (Contradiction - We can't have AuA and ~AuA.)
A6: ~(~Gr) + ~(~Ap) + Ap + AuA -> Ap => (Gregory and April did break the window.)

Both A1 and A6 are valid solutions. Altough we still don't know if August and June may have participated in breaking the window, or if we accept A1, it may be the case that none of the four broke the window. There is no premise stating that one or more of the four broke the window. Instead of choosing an answer, we go ahead and ask the four again:

Question B: Who broke the window?

GrB:  Ju              (June was)
ApB:  ~Au             (August was not)
AuB:  ~Ap             (April was not)
JuB:  ~GrA -> ~(~Gr)  (GrA is false, therefore Gregory is)

Taking the previous truth table. For Ax when x may be 1 or 6:

B   GrA         ApA         AuA         JuA                   GrB   ApB   AuB   JuB             Conclusion  
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------  
A1  ~Gr {T}     ~Ap {T}     ~(Ap) {F}   ~(AuA) -> ~(Ap) {F}   Ju    ~Au   ~Ap   ~GrA -> ~(~Gr)  =>  ~Gr   ~Ap   ~Ap   ~Ap   Ju  ~Au   ~Ap   Gr  ->  Contradiction (We can't have GrA and ~GrA)  
A6  ~(~Gr) {F}  ~(~Ap) {F}  Ap {T}      AuA -> Ap {T}         Ju    ~Au   ~Ap   ~GrA -> ~(~Gr)  =>  Gr    Ap    Ap    Ap    Ju  ~Au   ~Ap   Gr  ->  Contradiction (AuA and AuB are contradictory)  

Assuming all the answers for question B are true, then neither A1 or A6 are valid because there's a contradiction. So let's find another Ax where All B answers don't reach a contradiction:

B   GrA           ApA           AuA         JuA                   GrB   ApB   AuB   JuB             Conclusion
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
A2  ~Gr {T}       ~(~Ap) {F}    Ap {T}      ~(AuA) -> ~(Ap) {F}   Ju    ~Au   ~Ap   ~GrA -> ~(~Gr)  =>  ~Gr   Ap    Ap    ~Ap   Ju  ~Au   ~Ap   Gr  ->  Contradiction (We can't have AuA and ~AuA or GrA and ~GrA)
A3  ~Gr {T}       ~(~Ap) {F}    ~(Ap) {F}   AuA -> Ap {T}         Ju    ~Au   ~Ap   ~GrA -> ~(~Gr)  =>  ~Gr   Ap    ~Ap   Ap    Ju  ~Au   ~Ap   Gr  ->  Contradiction (We can't have GrA and ~GrA)
A4  ~(~Gr) {F}    ~Ap {T}       Ap {T}      ~(AuA) -> ~(Ap) {F}   Ju    ~Au   ~Ap   ~GrA -> ~(~Gr)  =>  Gr    ~Ap   Ap    ~Ap   Ju  ~Au   ~Ap   Gr  ->  Contradiction (We can't have AuA and ~AuA)
A5  ~(~Gr) {F}    ~Ap {T}       ~(Ap) {F}   AuA -> Ap {T}         Ju    ~Au   ~Ap   ~GrA -> ~(~Gr)  =>  Gr    ~Ap   ~Ap   Ap    Ju  ~Au   ~Ap   Gr  ->  Contradiction (We can't have AuA and ~AuA)

Every possibility is a contradiction, therefore we can't assume all answers to question B are true. Someone must be lying to question B. So let's make a truth table for question B:

Truth table for question B

B   GrB   ApB   AuB   JuB
--------------------------
1   T     T     T     T
2   T     T     T     F
3   T     T     F     T
4   T     F     T     T
5   F     T     T     T
6   T     T     F     F
7   T     F     T     F
8   T     F     F     T
9   F     T     T     F
10  F     T     F     T
11  F     F     T     T
12  T     F     F     F
13  F     T     F     F
14  F     F     T     F
15  F     F     F     T
16  F     F     F     F

So if we try to match all these with the question A truth table we get a bigger truth table. Ruling out the contradictions where we can't have AuA and ~AuA (AuA and JuA can't have different boolean values) or GrA and ~GrA (GrA and JuB can't have the same boolean value), also ruling out the cases where all answers to question B are true (B1) which we already know are invalid, we are left with the following set:

A+B     GrA   ApA   AuA   JuA   GrB   ApB   AuB   JuB
------------------------------------------------------
A1+B2   T     T     F     F     T     T     T     F
A1+B6   T     T     F     F     T     T     F     F
A1+B7   T     T     F     F     T     F     T     F
A1+B9   T     T     F     F     F     T     T     F
A1+B12  T     T     F     F     T     F     F     F
A1+B13  T     T     F     F     F     T     F     F
A1+B14  T     T     F     F     F     F     T     F
A1+B16  T     T     F     F     F     F     F     F
A6+B3   F     F     T     T     T     T     F     T
A6+B4   F     F     T     T     T     F     T     T
A6+B5   F     F     T     T     F     T     T     T
A6+B8   F     F     T     T     T     F     F     T
A6+B10  F     F     T     T     F     T     F     T
A6+B11  F     F     T     T     F     F     T     T
A6+B15  F     F     T     T     F     F     F     T

This table allows us to see that after all contradictions ruled out, we essentially narrowed down to the combination of GrB, ApB and AuB boolean values in two set of circumstances with their own premises. Let's call them C and D:


Possibilities

Circumnstance C (A1)

Gregory did not break the window. (GrA {T} and JuB {F})
April did not break the window. (ApA {T}, AuA {F}, JuA {F})

Circumnstance D (A2)

Gregory did break the window. (GrA {F} and JuB {T})
April did break the window. (ApA {F}, AuA {T}, JuA {T})

We are now exposed to another contradiction from the set we made earlier. If AuB is true while ApA is false or vice versa, then the conclusion is invalid. So let's eliminate those cases:

A+B     GrA   ApA   AuA   JuA   GrB   ApB   AuB   JuB
------------------------------------------------------
A1+B2   T     T     F     F     T     T     T     F
A1+B7   T     T     F     F     T     F     T     F
A1+B9   T     T     F     F     F     T     T     F
A1+B14  T     T     F     F     F     F     T     F
A6+B3   F     F     T     T     T     T     F     T
A6+B8   F     F     T     T     T     F     F     T
A6+B10  F     F     T     T     F     T     F     T
A6+B15  F     F     T     T     F     F     F     T

This is the formal(ish) version of the arguments after the truth values are applied, simplified to omit the information we already take for granted:

A1+B2 Ju {T} ~Au {T} => Ju ~Au -> Ju
A1+B7 Ju {T} ~(~Au) {F} => Ju Au -> Ju + Au
A1+B9 ~(Ju) {F} ~Au {T} => ~Ju ~Au -> None
A1+B14 ~(Ju) {F} ~(~Au) {F} => ~Ju Au -> Au
A6+B3 Ju {T} ~Au {T} => Ju ~Au -> (Gr + Ap) + Ju
A6+B8 Ju {T} ~(~Au) {F} => Ju Au -> (Gr + Ap) + Ju + Au
A6+B10 ~(Ju) {F} ~Au {T} => ~Ju ~Au -> (Gr + Ap)
A6+B15 ~(Ju) {F} ~(~Au) {F} => ~Ju Au -> (Gr + Ap) + Au


Final conclusion

So in the end of it all it's really a truth table from the answers of Gregory and April to the second question (Gregory: It was June and April: August didn’t do it)

Which gives eight possibilities without any contradiction depending on which combination of these two sentences we consider true or false.

There is one possibility where the four broke the window (Gregory, April, August, June).

There are two cases where three of the four break the window, in which Gregory and April are always present and they may break the window either with August or June.

There are two cases where two of the four break the window (Gregory and April or August and June).

There are two cases where only one person broke the window (June or August).

There is one possibility where no one of the four broke the window.

This covers the logic part of the answer and should be a valid answer giving the information available.


Assumptions

Now if we assume two premises not given by the OP:

Premises 3 and 4: There is one and only one person in the set Gregory, April, August, June who broke the window.

Then we'll have to narrow down even further and take other measures not related to logic alone to choose a culprit between June and August.

In this case, I would judge this way:

The first thing is that I have only what I called "Circumstance C" before. Which means The answers to the first question must have these truth values:

Gregory: I was not! (True)
April : I didn’t do it. (True)
August: April was. (False)
June : August says the truth (False)

Now we have two choices:

If June lied in the second answer, then June broke the window.
If everyone but August lied in the second answer, then August broke the window.

Here are the sentences again:

  • Gregory: It was June
  • April: August didn’t do it
  • August: April didn’t do it!
  • June: Gregory was lying.

Choice one: If June is the culprit
Gregory says it was not Gregory and said it was June, April says it was not April or August, August lied about it being April but now admits it wasn't April. June lied that August was telling the truth about it being April, and now lies that Gregory was lying about not being Gregory. The one who broke the window is June.

Choice two: If August is the culprit
Gregory says it was not Gregory but now is lying that is was June. April said it was not April but now is lying that it was not August. August lied about it being April but now admits it wasn't April. June lied that August was telling the truth about it being April, and now lies that Gregory was lying about not being Gregory. The one who broke the window is August.

My judgement

In both cases June lied two times. Even if June didn't lie about June itself, this is suspicious behaviour. Also the first June lie was supporting a lie from August, this is suspicious behaviour too. And finally the second June lie is calling a liar on the one who accused June.

So I would isolate June and confront those facts until June either tell the truth or give new information. If I couldn't come to a conclusion and had to choose someone, it would be June.

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    $\begingroup$ I marvel at the effort you put into this answer. Awesome! $\endgroup$ Aug 11 '20 at 23:00
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From the first set of statements:

Gregory: I was not!
April : I didn’t do it.
August: April was.
June : August says the truth

We know that two are true and two are lies. Now

August's and June's statements are either both true or both lies, since June's statement is about August's statement.

  • If they're both true, then

    April is the culprit, therefore April's statement is false but Gregory's statement is true, contradiction.

  • If they're both lies, then

    Gregory and April are both telling the truth, therefore both innocent.


From the second set of statements:

Gregory: It was June
April: August didn’t do it
August: April didn’t do it!
June: Gregory was lying.

Assuming that each person is consistent, either telling the truth both times or lying both times, then

Gregory and April are again telling the truth, August and June are lying. But that means April and June are both the culprits!

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  • $\begingroup$ If Agust are lying we can see the April says “April didn’t do it!” (April telling the truth) So the answere is June $\endgroup$
    – user36514
    Aug 10 '20 at 8:12
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    $\begingroup$ @user36514 Instead of giving away the answer in the comments, please consider clarifying the question as requested, so that we can arrive at the correct answer ourselves. $\endgroup$
    – F1Krazy
    Aug 10 '20 at 8:34
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The first liar is:

August - We can see he's contradicting himself in the two scenarios.

The second liar is:

June - She affirms that August is telling the truth even though he's clearly not.

So the ones telling the truth are:

Gregory and April

The one to blame is:

June - Since April just affirms that she wasn't the one to do it nor August, and Gregory (who's telling the truth) blames June.

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First Statements:

Gregory: I was not!
April : I didn’t do it.
August: April was.
June : August says the truth

Given, two of them said truth, and two of them lied. There are only two possibilties, so a person can say either truth or false(lie).

Choice - 1

Considering June as Truth:
As June is truth, August says the truth too (we have got two of them who says truth: June, August So Gregory and June must be lying), and according to August, April is the culprit, which is staisfied as April lied, But even Gregory was lying, so he must be the culprit too But as there only one culprit involved, So Considering june said truth is False

Choice - 2

Considering June as Lied :
As June lied, By June words, August lied too (So we got two of them who lied :June, August), So Gregory and April says the truth, Considering that Gregory and April are not Culprit

Second Statements:

Gregory: It was June
April: August didn’t do it
August: April didn’t do it!
June: Gregory was lying.

As Gregory and April didn't do it, then it must either be August or June. Asking again changed the August statment into truth, and April says August didn't do it, So turns out that June is the Cuprit not only as he is the only option left but also he defends himself that Gregory is lying whereas Gregory has no reason to lie as he didn't do it.

Culprit is

June

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There is information missing here for the second set of answers, but we can make assumptions that do lead to a solution. However, we also are missing if there can be more than 1 culprit.

Let's start with an important note on the missing info:

We know that August provides conflicting answers, so the missing information cannot be that a liar always lies: August MUST be lying once and telling the truth once. As such, the only way missing information can lead to a conclusion is if the missing info is 'again, two lied and two spoke the truth'. It might be 'this time, X lied and 4-X spoke the truth', but then that would be too obvious to not leave out. Leaving it out by accident suggests that the second set of answers shares a characteristic with the first, and we already know it cannot be liars-always-lie. There is 1 other option: 'This time, those who lied before spoke the truth, and vice versa'.

Now, the first set of answers:

August and June overlap, either both speak the truth or both lie. This leads to two possible scenarios. Scenario A: August and June are speaking the truth: April is a culprit. This means Gregory also lied, meaning April and Gregory are both culprits. Scenario B: August and June both lie. This means Gregory and April are telling the truth, so Gregory and April are both innocent.

Then, we get to the second set of answers.

We assume that 2 lie and 2 tell the truth. But we cannot tell who based on the first set. Gregory and June clash, so 1 of them lies and 1 of them is telling the truth. Next, either April or August has to be a liar, meaning one of them is a culprit and the other is innocent. Under scenario B, we know April was innocent. This means August is telling the truth, and April is lying. This means August is a culprit. IF we only have 1 culprit, then that means Gregory is lying now and only August did it. However, if multiple culprits are allowed, then Gregory could be lying or telling the truth, so we don't know if June ALSO did it. Unless of course '1 lie, 1 truth, per person' applies, then we know Gregory lies this time around and June is innocent. Under scenario A, we know April and Gregory are both culprits. This means August lies this time around, April is telling the truth, August is innocent. Again we cannot resolve June vs Gregory, so we don't know if we have 2 or 3 culprits. If, however, '1 lie, 1 truth, per person' applies, then we know this time Gregory is telling the truth and June is lying, so we have 3 culprits.

So, based on '1 culprit mandatory, or multiple culprits possible' and '2 lies + 2 truths second time' or 'who lies and who tells the truth flipped in the second set of answers', we may or may not get an answer with 1, 2 or 3 culprits. So we cannot tell the culprit for sure, even with assumptions of the missing information. But likely the intent was:

1 culprit, second time again 2 lies + 2 truths, so August is the culprit.

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