Who is the murderer?

Look at the first and last statements. They make contradictory claims, so one of them must be true and the other one false.

We are told that there is only one true statement, so the middle statement must be false. (Otherwise there would be two true statements).

The middle statement says that "Joe is not the murderer". Since we have proved that this statement is false we conclude that

$\boxed{\text{Joe is the murderer}}$

While we could conclude a culprit quickly by virtue of the first and last statement being contradictions, we should prove that the problem and its answers are valid, by showing that only one of the answers is correct. So we need to check that there aren't multiple cases where there are two lies and one truth.

Case study for each person being the murderer - Where 1 truth and 2 lies confirms a murderer.

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Did the cops arrest all the wrong suspects? (I.e. It was someone else - not Alex, Jenifer, or Joe)

It wasn't someone else.

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Is Alex the murderer?

It's not Alex.

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Is Jenifer the murderer?

It's not Jenifer.

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Is Joe the murderer?

As there are no other cases with 1 truth and 2 lies.

$\large\boxed{\color{#D61F06}Joe}$ is the murderer!

Assume Alex's statement is true. Then Jennifer is the murderer and her statement is false which means that joe is the murderer aswell. $\boxed{Contradiction}$ . Now, assume Jenifer's statement is true. Then Joe is lying and Jenifer is the murderer which makes Alex's statement true aswell. $\boxed{Contradiction}$ . Hence, Joe is telling the truth and Jenifer is lying which leads us to say that the murderer is $\boxed{Joe}$

We know all three statements,

Alex: "Jenifer is the murderer."

Jenifer: "Joe is not the murderer."

Joe: "Jenifer is not the murderer."

Now, if Alex killed Robert,

• Alex's statement is false , Jenifer and Joe's statement is true . Since only one statement is true and other two are false, so Alex isn't the killer

If Jenifer killed Robert,

• Alex and Jenifer's statement is true , Joe's statement is false. So, Jenifer isn't the killer

If Joe killed Robert ,

• Alex and Jenifer's statement is false , Joe's statement is true. So Joe is the killer

Therefore , Joe murdered Robert.

The Butler did it! But in all seriousness if you take a look at each statement and pick one to be true, you know the other two are false, if any false statement says contradict each other, or the true statement, then the statement you picked to be true, isn’t true. If you pick the statement “Jenifer is the murderer”, then the two other statements are lies. The opposite of those two sentences is the truth. So the truth would be “Joe is the murderer” and “Jenifer is the murderer”. Those statements contradict each other. But if you choose “Jenifer is not the murderer” then the opposite of the two lies would be “Joe is the murderer” and “Jenifer is not the murderer”. None of those statements contradict, therefore they are accurate and Joe is the murderer

If Alex says Truth then consequently the statement of Jennifer becomes also true.
If Jennifer says Truth then there is a contrast between the statements​ of Alex and Joe i.e. one says Jennifer is the murderer and another says Jennifer is not the muderer. so Jennifer is not true.
Now, if Joe is true then Alex and Joe's statements​ becomes same. And from Jennifer's statement we get that Joe is the murderer

Alex and Jennifer are both giving clean chit to Joe, and since two can not be telling the truth, so both are lying and consequently Joe is the culprit.

Why alex killed him?