Solving A Murder Case

An easy way to find the murderer is analyzing statements 2 and 3. Both show the same result. So, both statements are either true or false. But as no two statements can be true, so both must be false so the Statement 1 (X's) is true and "Y" is the murderer.

Suposing that the Z's statement is true and that Y and X's statement is false, Z is the murderer. But looking at Y's statement we can conclude that Z's statement is false. Suposing that the Y's statement is true and that X and Z's statement is false, Z is murderer. But looking at Z's statement we can conclude that Y's statement is false. Assuming that the X's statement is true and that Y and Z's statement is false, Y is murderer. Looking at other statement we can conclude that X's statement is true and Y is murderer.

1st consider "x " statement is true. After that "y" and "z". Make table between statements(columns) vs murderers(row)
For murderer X------F F F statements For murderer Y----- T F F
For murderer Z------F T T
Our condition is only one statement is true So "Y" is the murderer

one statement not necessarily be true ; one situation has to be true(either x,y or z was the killer) both statement 2 and 3 point out that z is the killer since they both point out to a single situation,so i see no reason to dismiss statement 2 and 3

If statement of Y is true then the statement of Z is also true, but there only one statement gotta be True, Which the Statement of X !

Am I the only one who found that X is saying the thruth because if he wanted to lie he could just have said that Z is the murderer, like that each statement is saying that Z is the murderer so it would be solved and Z would end up in jail. But no he is saying that it's Y so he is saying the truth. Is that logical ? Im not sure lol.

If what Z says is true, then Z is the murderer. But then what Y says must be false and Z cannot be the murderer. Therefore Z is telling false. Therefor Y is telling false and X the truth

Look at statements 2 and 3. Both indicates "Z" as murderer. So, both statements are either true or false. But as no two statements can be true, so they both must be false leaving beyond Statement 1 which is true. So Y is the murderer.

The problem can be solved using logic formulations as well (although I would like some feedback on this one): Consider: X - X committed the murder, Y- Y committed the murder, Z - Z committed the murder.

So, rule 1: $Y\overline{Z}\overline({\overline{X+Y}})$ \ rule 2: $\overline{Y}Z(\overline{\overline{X+Y}})$ \ rule 3: $\overline{Y}\overline{Z}(\overline{X+Y}) = \overline{Y}\overline{Z}(\overline{X}.\overline{Y})$ \

Since only one of these is true, we can connect them by disjunction (OR)

rule 1 OR rule 2 OR rule 3 = $Y\overline{Z}\overline{\overline{X+Y}} + \overline{Y}Z\overline{\overline{X+Y}} + \overline{Y}\overline{Z}(\overline{X}.\overline{Y})$ = $Y\overline{Z}X + Y\overline{Z} + \overline{Y}ZX + \overline{Y.Z.X}$ = $Y\overline{Z}$ (by absorption of the first two terms) + $\overline{Y}(X xnor Y)$ (by distributive law on the last two terms, and considering the formula for xnor)

Now, either of these can be true, which means: i) Y committed the murder, Z did not. ii) or, Y did not commit the murder and either X and Z both committed it, or neither did.

Given the problem, we know that only one of them committed the murder, so it can't be both. So the second term is essentially useless.

Which means, Y committed the murder.

Well, that's what I figured anyway! Feedbacks?

If X and Y are both lying, then neither Y nor Z are the murderer, however, if Z is telling the truth then either X or Y is the murderer, so why can't X be the murderer? I can understand the logic for Y being the murderer but doesn't this work too?