$3$ people in a room: one tells lies, one tells truth, one tells a truth or lie at random

Question

So I am stuck on this question and I am not really looking for a solution as such more some advice on how to approach a problem like this:

Three people called Alf, Beth, and Gemma, sit together in the same room.

One of them always tells the truth.

One of them always tells a lie.

The other one tells truth or lies at random.

In each of the following situations, your task is determine how each person acts.

Suppose that Alf says “I always tell lies” and Beth says “Yes, that’s true, Alf always tells lies”.

Who always tells the truth? Who always lies? Briefly explain your answer.

One approach that I am trying to take is by forming all the ways in which they could either tell the truth, lie or random like so.

T=always telling the truth

L=always telling a lie

R=one tells truth or lies at random

So my permutation are as follows

• TRL
• TLR
• RTL
• RLT
• LTR
• LRT

But now this is where my problem lie because of the way it worded as in

Suppose that Alf says “I always tell lies” and Beth says “Yes, that’s true, Alf always tells lies”

I am not sure I am applying my permutation in the correct way, so as an example, taking TRL

so to me this combination dose not work because Alf admits that he always lie then TRL wont apply.

But then I am thinking he could be telling the truth by saying he always lying he telling the truth. So the two contradict each other.

which confuse me and I am just wondering is there an alternate way of approaching this type of question?

Answer 1

Using the given permutations and using TLR and case 1, we can ask the question 'Is Alf the truth teller?'.

The answer to this would be No. The reason being that when he say ''I always tell lies'' this statement contradicts because you cannot tell the truth, if every word you say is a lie.

So this rules out

• TRL
• TLR

So now using RTL as the next permutation to analyse. So now we ask the question 'Is Alf telling lies or truth randomly?'

The answer would be Yes.

The reason being is that as before he cannot be telling the truth and he cannot be a lair because if he lies his statement becomes the truth which as seen before he is not a truth teller.

So this leave the conclusion that he is either telling a truth or a lie randomly.

So this removes the permutations

• LTR
• LRT

And thus left with

• RLT
• RTL

Now the next question is, 'is Beth a lair or a truth teller'?

The answer is she a lair.

The reason for this is that as alf is telling lies some of the time and truth the other then her statement is incorrect in saying 'Alf always tells lies'.

So this leads to the conclusion that RLT is the correct permutation.

To concluded Alf tells lie and truth randomly, Beth is always lies and then Gamma must always be telling the truth.