Question: Elizabeth is engaged.
* Her fiancé is either Arthur, Barry, Colin or Derek.
* Each of the four men and Elizabeth either always tells the truth or always lies.
* Arthur says: "Exactly one of us four men always tells the truth."
* Barry says: "Exactly one of us four men always tells the lies."
* Colin says: "Arthur or Barry is Elizabeth's fiancé."
* Elizabeth says: "My fiancé and I either both always tell truth or both always lie."
Who is Elizabeth's fiancé?
Answer:
Barry is Elizabeth's fiancé.
Analyse the statements made by Colin and Elizabeth first.
From (6), if Elizabeth always tells the truth, then her fiancé always tells the truth and if Elizabeth always lies, then her fiancé always tells the truth. Hence, Elizabeth's fiancé always tells the truth.
Let's assume that statement made by Colin (5) is false. Then Colin lied and Derek is the truth-telling fiancé. But then statement made by Arthur can not be true (if true, then there are 2 truth-tellers which contradicts itself). And it can not be false either (if false, then there are at least 2 truth-tellers, which makes Barry's statement false and
hence there is just a one truth-teller i.e. Derek, again contradiction!!!). Hence statement made by Arthur contradicts itself. So the statement made by Colin is true i.e. Arthur or
Barry is Elizabeth's fiancé.
As statement made by Colin is true, the statement made by Arthur is false. Then because
Elizabeth's fiancé always tells the truth, Barry is Elizabeth's fiancé. Then the statement made by Barry is true and thus Derek always tells the truth.
Thus, Barry is Elizabeth's fiancé. .