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### Unique Relationship Puzzle

Unique Relationship Puzzle - 24 December

A man escapes from jail using help from his girlfriend. The investigators suspect four girls of being the man"s girlfriend.

Out of those four girls, one is his girlfriend who is lying. Two of the girls are completely innocent and are speaking the truth. One of the girls is the man"s sister who is helping the girlfriend lie. Following are the statements from all four of them:

Ariana: "Myrna is his girlfriend.
Jane: "Angelina is lying."
Angelina: "Ariana is lying."
Myrna: "Jane is not his sister."

Can you find out who is the man"s sister among them?

1. If Ariana is telling the truth, then Myrna is a liar.
If Myrna is lying, then Jane is his sister. If Jane is his sister, then she's lying about Angelina lying. If Angelina isn't lying, then Ariana is lying, which would contradict the fact that Ariana is telling the truth. That means Ariana is definitely lying and could be his girlfriend or sister.
So Angelina must be telling the truth since Ariana has to be lying. So Jane must also be lying, and Myrna is a truth-teller. Since Myrna is telling the truth, Jane isn't his sister and must be his girlfriend. So Ariana is the other liar, so she must be his sister.
Ariana is the man's sister.
P.s Nice logical problem (far above 3/5 difficulty, if I may say so...)

2. Ariana is the sister and Jane is the girlfriend.

Suppose if Ariana is his sister and we know that his statement is false.
This will mean that Myrna is not his girlfriend.
If Angelina is telling the truth, then Ariana must be lying that we have already assumed.

If Jane is his girlfriend, then she must have spoken a lie as well.
This will mean Angelina is speaking the truth which is true in what we have assumed.

At last, Myrna said that Jane is not his sister which is also true as we have made the assumption already that Jane is his girlfriend.

In what we have assumed, all the statements fit perfectly and the given conditions are satisfied. If you make any other assumption, you will find that one or more conditions are not fulfilled.