Discrete Mathematics - Logics Puzzles

[jhson114]

I ran into this logic puzzle and have been working on it for couple hours now but i can't seem to explain clearly why the answer i came up with is the answer. Heres the question:

The police have three suspects for the murder of Mr. Cooper: Smith, Jones, and Williams. Smith, Jones, and Williams each declare that they did not kill Cooper.
Smith also states that Cooper was a friend of Jones and that Williams disliked him. Jones also states that he did not know Cooper and that he was out of town the day Cooper was killed. Williams also states that he saw both Smith and Jones with Cooper the day of the killing and that either Smith or Jones must have killed him. Cna you determine who the murderer was if
a) one of the three men is guilty; the two innocent men are telling the truth, but the statements of the guilty man may or may not be true?
b) innocent men do not lie?

For (a)
Well, this is what i think (im not too good with p->q, this kinda stuff and can only say it in plain words). assuming smith is innocent and telling the truth, this means that jones is a friend of cooper and williams disliked cooper. this also means that jones ied by saying he does not know cooper and that he was not with cooper on the day of murder. Since two innocent men are telling the truth, and have determined that jones lied (assuming smith is telling the truth), then williams also must be telling the truth. This means that jones was with cooper and either jones or smith killed him. Since smith is innocent, then jones is guilty.

Asuuming that Mr wiliams is innocent and is telling the truth, then both smith and jones was with cooper and either smith or jones must have killed him. If this is the case, then jones lied by saying he was out of town on the day of murder and he did not know cooper. Then this makes smith's statement that jones was a friend of cooper. Therefore jones is the murderer.

For (b)
I said you cannot really determine for sure, because you can have multiple murderers. Just like case question a above, if there's only one murderer, then its jones; however the question doesn't explictly say that so there can me multiple murderers. This means that if jones is innocent, then both smith and williams is lying therefore they are both murderers. also all three can be lying that they all could be murderers.

Im not sure if I am thinking this correctly but this is what i came up with and if someone could verify this question for me i'd be really really thankful. thanks in advance.

 

[Andrew Mason]

The police have three suspects for the murder of Mr. Cooper: Smith, Jones, and Williams. Smith, Jones, and Williams each declare that they did not kill Cooper.
Smith also states that Cooper was a friend of Jones and that Williams disliked him. Jones also states that he did not know Cooper and that he was out of town the day Cooper was killed. Williams also states that he saw both Smith and Jones with Cooper the day of the killing and that either Smith or Jones must have killed him. Cna you determine who the murderer was if
a) one of the three men is guilty; the two innocent men are telling the truth, but the statements of the guilty man may or may not be true?
b) innocent men do not lie?

Your answer for a) is correct.
The answer for b) has to take into account what the suspects said. It also has to take into account the possibility that none are guilty. It also has to take into account the possibility that a witness can be mistaken rather than deliberately lying.

If innocent people do not lie then at least one of Smith and Jones is lying since their statements conflict - both cannot be true. Similarly, at least one of Williams and Jones must be lying because their statements conflict. There is no conflict between Smith and Williams, however.

If Smith is innocent then he is telling the truth and Jones must be lying. Jones' false denial of knowing Cooper means he is not innocent. This scenario is consistent with what Williams says. This does not mean that Williams is telling the truth, however. We just cannot conclude that he is lying. So at least Jones is the murderer. We cannot exclude the possibility that Williams was not in on it.

If Jones is innocent, then he is telling the truth and he was not with Cooper on the day of the murder. This means that both Williams and Smith are lying and both must, therefore, be the murderers.

If Williams is innocent, then he is telling the truth and Jones is lying, so at least Jones is the murderer. Again, Williams' statement is consistent with Smith's, so we cannot tell if Smith is lying. Smith could be in on it.

Of course, they all could be in on the murder and one of them telling the truth. It does not say that murderers necessarily lie.

And there is also the possibility that at least one of Smith, Jones and Williams is a woman and, therefore, could be innocent and still be lying.

AM