This problem has got me going in circles for several hours and I'm not sure what to do, the problem is as follows:
At a semiconductor laboratory in Hsinchu a security guard caught five
technicians accessing a high level security area reserved for the most
trusted scientists working in a new integrated circuit for an upcoming
computer. However the security film is not very clear and the security
team can only be sure that four out of five technicians have the
access key card to enter the chipset room.During interrogation the security team deduces that two of the
technicians are lying and the other are telling the truth.The answers given by the laboratory technicians were as follows:
Audrey: Gwendolyn does not have an access credential.
Dorothy: I was entrusted an access key.
Marina: Hannah has an access key.
Gwendolyn: Audrey is lying.
Hannah: Dorothy is telling the truth.
Based on this information. Which of the technicians does not have
access to the chipset room?
Typically I would provide something but in this case I'm stuck at the very beginning. So far the only thing that I found it is that such problem seems to be a Knights and knaves kind which is related to logic. But in this case there are five individuals, hence the number of possible combinations would mean $2^5=32$ which raises a flag to me as $32$ combinations seems too big to make it a practical approach to try one by one. Therefore I need help into finding a solution or a method which would ease and simplify or solve this problem easily.
Can somebody help me with this? I'm not very knowledgeable with this type of problems. It would help me a lot to visualize what's going on if the proposed solution would include some sort of table or grid so I could identify the concluded result.
Best Answer
With the new information (that exactly $2$ are lying) we can solve the problem.
There are $5$ states to consider according to whichever lacked access. We list them all:
If $A$ lacked access: $(F,T,T,T,T)$
If $D$ lacked access: $(F,F,T,T,F)$
If $M$ lacked access: $(F,T,T,T,T)$
If $G$ lacked access $(T,T,T,F,T)$
If $H$ lacked access: $(F,T,F, T,T)$
By inspection, Hannah is the guilty party.
To stress: we need some rule to let us know how to evaluate the various True-False configurations. A priori, we have no idea what they might mean. A natural rule would be, say, "the person who lacked access is lying, everyone else is telling the truth". That rule doesn't lead to a unique solution here (though you can narrow down the list of suspects to $A,G$. As it stands, we are given the rule "exactly two people are lying". Happily, that rule does lead to a unique solution.