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you are a detective in the local neighborhood. Four windows have been broken: the school's window, Mrs. Smith's kitchen window, Mr. Brown's car window, and the police officer's window. The four suspects are Alex, Collin, Josie and Raymond, and we know that each one of them broke a single window

Alex said:

I did not break Mrs. Smith's kitchen window, and I did not break Mr. Brown's car window.

Collin said:

Alex did not break the school's windows.

Josie said:

I know what happened. Collin broke Mrs. Smith's kitchen window and Alex broke the police office's window.

Raymond said:

I broke Mr. Brown's car window or the school's window and I am sorry.

You know that one of the suspects is a liar and three of them are telling the truth. Who broke Mrs. Smith's kitchen window?

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    $\begingroup$ Can you clarify: if someone says "A and B" and is a liar, is it possible that one of A,B is true and the other false, or should we make the assumption common in this sort of puzzle that a liar will lie about everything they say? $\endgroup$ – Gareth McCaughan Jan 13 '18 at 22:53
  • $\begingroup$ @GarethMcCaughan Following usual logic, a AND b is false iff either a is false OR b is false. $\endgroup$ – Mr. Xcoder Jan 13 '18 at 22:54
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    $\begingroup$ @Mr.Xcoder I know what usual logic says, but I have seen puzzles of this sort in which a different convention is used, hence the question. $\endgroup$ – Gareth McCaughan Jan 13 '18 at 22:54
  • $\begingroup$ Collin said: "Alex did not break the school's windows." Is the s a typo or is it the basis of the lateral-thinking tag. $\endgroup$ – ibrahim mahrir Jan 13 '18 at 23:24
  • $\begingroup$ The last bit by Raymond "and I am sorry" introduces ambiguity to the puzzle. He could be lying about that, which would lead to a different answer. $\endgroup$ – user3294068 Jan 31 '18 at 16:21
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First:

Collin is not the liar. Because if he is, there will be two liars: If Collin is a liar, then Alex must've broken the school's window, thus Josie is a liar too.

Second:

Alex is not a liar. Because if he is, then his statement is a lie, thus Alex must've broken either Mrs. Smith's kitchen window or Mr. Brown's car window. So Josie is a liar too. We can't have two liars, thus Alex is telling the truth.

Furthermore:

If Raymond is a liar, then he must've broken Mrs. Smith's kitchen window or the police office's window (the only other two possibilities), thus making Josie a liar too (he couldn't possibly broke one of those because Josie has confirmed that both windows were broken by different people). So Raymond is telling the truth.

And

Josie is the liar.


Let's recap:

Alex broke the police office's window (using his and Collin's statements).

And:

Collin did not break Mrs. Smith's kitchen window. (using the falsy half of Josie's statement, the half about Collin).

And:

Raymond did not break Mrs. Smith's kitchen window. (using his statement which we proved to be true).

So:

It was Josie who broke Mrs. Smith's kitchen window.

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  • $\begingroup$ If Josie is the liar it means Alex didn't broke Police officers car. Alex is saying truth so he didn't break Mrs Smith and Mr. Brown's windows. But also according to Collin Alex didn't even break the school window. This mean Alex didn't break any windows? $\endgroup$ – prog_SAHIL Jan 14 '18 at 15:21
  • $\begingroup$ @prog_SAHIL Josie said "A and B". If she is a liar then either "A is false and B is true", "A is true and B is false" or "both A and B are false". We know that "B is true" (based on Alex's and Collin's statements), so if Josie is a liar then "A is surely false". Hope that makes sense. :D $\endgroup$ – ibrahim mahrir Jan 14 '18 at 21:35
  • $\begingroup$ Josie also said "I know what happened". If this was false, then everything else Josie says is a guess and we have no way of assigning truth or lie value to it. $\endgroup$ – theonetruepath Jan 15 '18 at 3:37
  • $\begingroup$ @theonetruepath I think that and the "I am sorry" part of Raymond's statement are not as important as the rest of the statements. If not, then this will be beyond the scope of logical deductions. $\endgroup$ – ibrahim mahrir Jan 15 '18 at 12:19
  • $\begingroup$ It is also possible that Raymond broke the school's window and is not sorry about it, and is therefore lying. That means Josie told the truth, and Colin broke Smith's window. $\endgroup$ – user3294068 Jan 31 '18 at 16:20
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If Alex lies then

he broke either Smith's or Brown's window. If Josie tells the truth then Alex broke the police window. Both can't be true, but if Alex is the liar then they must be. So Alex is telling the truth, and broke either the school or the police window.

Similarly, if Collin lies

and Josie tells the truth then we have a contradiction; but if Collin is the liar then that must be the case. So Collin is telling the truth, and Alex broke not the school window but the police window.

If Josie is lying

(and hence Raymond telling the truth) then -- since we know Josie is right about Alex -- Collin must not have broken Smith's window. Neither did Collin break the police window, since Alex did; so he broke either Brown's or the school's. Since Raymond is telling the truth, he broke the other one of those. Then Alex broke the police window and hence Josie broke Smith's window. This all seems consistent.

The other possibility is that Josie is telling the truth and Raymond is lying.

In that case (from Josie's testimony) Collin broke Smith's window. So Josie and Raymond, between them, broke Brown's and the school's windows, which means that Raymond is telling the truth, contradiction.

So the only possibility is

the one in the third paragraph, and it's Josie who broke Smith's window.

Note: this assumes

it isn't possible that Raymond did break Brown's or the school's window but isn't sorry and is therefore lying. That seems like a bit of a weak point, but without that assumption the question doesn't have a unique answer.

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If the liar is Alex, then: Alex must have broken Mrs. Smith's kitchen window or Mr. Brown's car window. In this case Josie must lie too (Alex broke the police office's window). So this scenario yields a contradiction.

If the liar is Collin, then: Alex must have broken the school's window, and Josie would lie too (Alex broke the police office's window.). Again, contradiction.

If the liar is Josie, then, assuming the false statement is Collin broke Mrs. Smith's kitchen window: Collin did not break Mrs. Smith kitchen window, but neither did Alex. That leaves us with two choices for Mrs. Smith's window: either Josie or Raymond. But Raymond said that he did break Mr. Brown's car window or the school's window, so Josie must have broken Mrs. Smith's window.

And therefore the answer is:

Josie broke Smith's window. Because if Josie lies, the scenario is logically consistent (assuming the statement we picked earlier is false).

Using more "formal" logic (well, not quite :P):

We know that:

  • (1) $\exists!\text{ liar}$
  • (2) $\forall \text{ windows }\exists!\text{ suspect}$
If Alex is not telling the truth, then the statement $$S_1=\text{(Alex didn't break Mrs. Smith's window) }\land\text{ (Alex didn't break Mr. Brown's car window)}$$ must be false. But that cannot be the case, since Josie also states $$S_2 = \text{(Alex broke the police office's window)}$$but $(1), (2)\implies\lnot\:S_1\land S_2$ cannot be true. Contradiction!

If Collin is lying, then $$S_1\land ¬ \text{ (Alex did not break the school's window)}$$implies that $S_3=\text{Alex broke the police office's window}$ must be true. But then we have a problem with$$S_4=\text{(Collin broke Mrs. Smith's window)} \land \text{(Alex broke the police office's window)}$$That is, $(1), (2)\implies S_4\land S_3$ cannot be true. Contradiction!

If the liar is Josie, then $S_4$ is false. Therefore either of its sub-statements must be false. I'll call them $S_{41}$ and $S_{42}$. Considering the first case, $S_{41}$ is false, then Collin didn't break Mrs. Smith. But from $(1)\land S_1$ we know that neither did Alex, and from $$(1)\land S_5 \:\text{(Raymond broke Mr. Brown's car window or the school's window)}$$we know that neither did Raymond. In this case, we can imply from $(2)$ that Josie did break Mrs. Smith's kitchen window. Great!

Now, for completeness' sake, if $S_{42}$ would have been false, $$\lnot S_{42}\land\text{(Alex did not break the school's window)}\land S_1$$must be false, since we know $(2)$ is true, and in this case Alex wouldn't break any window.

Assumptions

  • The puzzle has a unique solution, and therefore there is no need to analyse the fourth case, Raymond is lying.
  • The s in school’s windows is just a typographical error
  • A AND B” is false has its natural meaning, that either A is false OR B is false (or both)

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  • $\begingroup$ Consider rechecking: "Assume the false is Alex broke the police office's, then: Alex must have broken the police office's window". $\endgroup$ – ibrahim mahrir Jan 13 '18 at 23:41
  • $\begingroup$ @ibrahimmahrir Ugh, thanks for spotting that! Fixed $\endgroup$ – Mr. Xcoder Jan 13 '18 at 23:42
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Alex:

If Alex is lying then Josie must also be lying, which is a contradiction. Therefore Alex is telling the truth.

Collin:

If Collin is lying then Josie must also be lying, which is a contradiction. Therefore Collin is telling the truth.

Now we are left with two situations:

Either Josie is lying or Raymond is lying. Both have given compound statements so we are left with a choice.

If Josie is lying it must be about who broke Mrs. Smiths window, as we know that Alex must have broken the police officers window if Alex and Collin are telling the truth. If we assume that Raymond is telling the truth then he didn't break Mrs. Smiths window, and neither did Alex or Collin, so it must have been Josie herself.

The other option is that Raymond is lying. But if Raymond is lying about which window he broke then Josie must be lying as well, which is a contradiction. So Raymond is not lying about which window he broke, but rather about him being sorry.

Both of these solutions are logically valid, so we must find another basis on which to decide. Raymond gave a relatively unhelpful answer, lacking in specificity, leading me to doubt his intentions are genuine. This suggests that he is not really sorry for what he has done. I conclude that Raymond is the liar, leaving Collin as the person who broke Mrs. Smiths window.

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I would like to answer this, in terms of interpretation, and not using the heavy logic implied above, though they have better credentials and are absolutely correct in applying the logic necessary and then deducing.

Let's assume:

All of them are speaking are speaking the truth. If that is the case, and one of them is lying(Alex being considered the liar here) then Alex is breaking two or probably more windows, which defies the given constraints. So Alex is telling the truth about him not breaking the two windows.

Next up:

If what Collin says is a lie, then Alex must have broken the school's windows and let's keep that in mind. But this still doesn't necessarily have to be correct.

Now when:

Josie says, that Collin broke the kitchen window and Alex broke the police office's window, making Collin the alleged guy who broke the window. We'll come to him lying later in the answer.

But:

When Raymond says that he broke either of the two windows, if he is lying, then he either broke the police office's window or the Kitchen's window, which makes Josie a liar as well. This results in two liars which cannot be.

Now let's retrace our paths from here:

If Raymond is lying, then Josie is lying as well. If Collin is lying then Alex is lying as well. And if Alex is lying then Raymond is lying as well. This leads to us having two liars which cannot be the case. So it's definitely Josie lying, since him lying would mean, that all the others are necessarily telling the truth, making JOSIE THE LIAR AND THE ONE BREAKING MRS. SMITH'S KITCHEN WINDOW

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