# Cora's Death Logic Puzzle

When Cora was killed, Anna and Beth were questioned by the police about the manner of her death by poison.

Anna: If it was a murder, Beth did it.
Beth: If it was not suicide, it was murder.

The policemen made the following assumptions:

$1$. If neither Anna nor Beth lied, it was an accident.
$2$. If either Anna or Beth lied, it was not an accident.

Subsequent development revealed these assumptions were correct.

What was the manner of Cora's death: accident suicide or murder?

It was murder.

Assumption 1 (both tell truth = accident) contradicts Beth (if not suicide then murder). Therefore at least one of them wasn't telling the truth, and so (because of assumption 2) it wasn't an accident.

Now that we know it wasn't an accident, we also know that Beth is telling the truth (assuming only possibilities are suicide, accident or murder). So Anna is lying.

The only way for Anna to be lying is if it was a murder and Beth didn't do it.

I assume that Anna is guilty of murder, but I only have enough evidence to say it was murder and that Beth didn't do it.

• Actually, another way for Anna to be lying is if she in fact knew that Beth was surely innocent having personal proof of Beth's alibi. In which case, suicide is a valid possibility. Example scenario: Anna and Beth are sisters, who both love their mother Cora and hate each other, mutually believing the other to have maintained a toxic relationship with Cora. Each is naturally willing to implicate the other as a suspect. Cora, who loved both daughters was unable to handle the bitter rivalry of her children, and convinced of her failure as a parent, committed suicide. No conclusive answer. Feb 25, 2015 at 5:11
• @Chris Anna's statements is a standard logical If A, then B statement. For these types of statements, if A is false then the statement is, by definition, true. So if Cora committed suicide, it wasn't murder, and Anna is telling the truth. Feb 25, 2015 at 13:18
• flip a coin. If it's heads, I'm the president of the world. It's possible for Anna to know the certainty of Beth's innocence (and lie about it) without knowing whether or not the death was actually a murder. Feb 25, 2015 at 17:28
• @Chris Anna cannot be lying if it isn't a murder. If it isn't a murder, the then portion of the if-then statement is moot. If the if portion is false, by definition, the statement is considered true. Feb 25, 2015 at 17:35
• We don't know that what Anna said was false, merely that she lied. To understand the difference, please note that you should be able to determine whether or not the following statement is a lie without actually flipping a coin: "If it's heads, I'm the president of the world." Feb 25, 2015 at 17:43