# Eye exam for the truth-teller, liar and joker

A, B and C went for a eye exam together for their driver licenses. All 3 of them have bad eye sight and needed their own glasses to see clearly. However only 1 of them remembers to bring the glasses. During the exam, they pass around this pair of glasses and said the following while wearing the glasses:

A: The first letter is an E!
B: A is lying!
C: I can't see!

• One of them always tell the truth, one of them always lies, and the last one is a joker (either lies or tell the truth).
• The truth teller HAS to be able to see to tell the truth.
• We do not know if the first letter is an E.
• And of course, we do not know who brought his glasses.

With such obvious cheating the examiner was to fail them all. However the examiner has to let them pass if he cannot verify whose glasses belongs to.

Is there a way / ways they can pass the exam?

I can provide an example if needed.

Hint / example 1:

Say A is the liar, B is the joker and C is the truth-teller. The examiner can claims the glasses belongs to A. A can see the letter but lies about it. B is the joker anyways and C is telling the truth that he cannot see, thus failing them. Examiner can also claims the glasses belongs to B but I'll let you think about it.

• Is there a way to ascertain the truth-value of I can't see! ? Commented Mar 28, 2018 at 17:05
• Can we ask the patients as many questions as we want? Commented Mar 28, 2018 at 17:06
• @ABcDexter sure! "I can't see" refers to the person wearing the wrong glasses. Is that what you're asking?
– Alex
Commented Mar 28, 2018 at 17:15
• @Jordan.J.D Nope, just to avoid examiner go asking each of them "is this your glasses" type of work-around heh
– Alex
Commented Mar 28, 2018 at 17:16
• @Abigail Thanks for clarifying these points - a) The truth teller always tell the truth - so if he did not bring the glasses, he let the examiner knows. b) It does not matter if A, B and C knows but yes exactly one of them for each type.
– Alex
Commented Mar 28, 2018 at 22:09

All three people will pass.

This is fairly easy to prove. But first, we must prove who the glasses belong to. To do that, let's first remember this important point (which I shall call Important Point Alpha):

We were given three statements, one each from person A, B, and C, so we know that the truth-teller, the liar, and the joker each made one statement; we just don't know which statement was made by which person.

It's also important to keep in mind:

Axiom 1 of the puzzle states that the truth-teller always tells the truth, the liar always lies, and the joker can do either.

And finally, we must also pay special attention to:

Axiom 2 of the puzzle, which specifies that the truth teller HAS to be able to see in order to tell the truth.

And therefore, putting it all together:

1. We know from Important Point Alpha that the truth teller (whoever they are) definitely made a statement.

2. We know by the puzzle's Axiom 1 that the truth-teller's statement is the truth.

3. We know by the puzzle's Axiom 2 that the truth-teller's truth may only be spoken when wearing their own glasses.

Therefore, the glasses belong to the truth-teller. They cannot belong to anyone else without violating one of the axioms of the puzzle.

Okay, I'm convinced by your sexy display of pure logic, but which of the three people is the truth-teller?

So even though we know that the glasses belong the truth-teller, we don't know which of the three people that is. And in fact, we really can't know; the truth-teller could be any of the three, without causing a contradiction in the statements. If the truth-teller is A, then B is lying and C is the joker telling the truth. If the truth-teller is B, then A is lying and C is the joker telling the truth.

But my favourite case would be if the truth-teller was C; his glasses are sunglasses, and the testing room is pretty dark, so he can't see the test even with his own glasses on. In that case, then A is the joker lying (as they cannot see the test but claim that they can) and B is the liar (because they can't see to tell whether or not the joker lied but claim that they can), or A is the liar (cannot see the test but claims that they can) and B is the joker telling the truth (because they know who the liar is and know that the liar must have lied, even though they can't actually see it)

No way to tell which is which, so.. we have to pass them all, and let them receive their drivers' licenses.

Maybe it'd be a good idea to stay off the roads for a while.

• lol at the creativity with sunglasses! but sorry it's not the case - the question stated that "needed their own glasses to see clearly", which means if the glasses belongs to that person, he will be able to see.
– Alex
Commented Apr 2, 2018 at 17:36
• I added an example on the question - see if that clarify a thing or two~
– Alex
Commented Apr 2, 2018 at 17:46

Assuming the examiner knows the first letter, I think he can figure out who has the glasses.
We also assume that the truth teller has their glasses and can see.

Scenario 1 (First letter is E):

C can't be the truth teller because the truth teller can see and so could not say "I can't see!"
If B is the truth teller, then A is lying, but in this case the first letter is E, so A is not lying, therefore B is not the truth teller.
So we know A is the truth teller with the glasses, B is the liar, and C is the joker, who in this case is telling the truth.

Scenario 2 (First letter is not E):

C can't be the truth teller with glasses for the same reason as scenario 1.
A can't be the truth teller because their statement is false.
So B is the truth teller with glasses, A is the liar, and C is again the joker, telling the truth.

Conclusion

So as long as the examiner knows the first letter, he can figure out who the glasses belong to and fail them.

• Good attempt, however on the first line on Scenario 1 - if the glasses does not belongs to C then he is telling the truth that he can't see.
– Alex
Commented Mar 28, 2018 at 17:44
• Ok, then the line "The truth teller HAS to be able to see to tell the truth" is confusing to me. Do you mean that only the truth teller can make a correct statement about a letter they claim to see? Commented Mar 28, 2018 at 17:57
• @couriouserd I think that statement is to assert that the truth teller cannot name a letter if they cannot see. A truth-teller could still make a true statement about other things. The joker can make a true statement about a letter they see (or a false statement if they saw the letter), or could SAY they see a letter when they cannot see at all, as this would also be a lie. And a liar could SAY they see a letter if they cannot see at all. Commented Mar 28, 2018 at 18:02

The examiner will be able to fail all three of them

To fail them, the examiner must be able to find out whose glasses they are. By looking at the three statements the first thing we can see is that:

The glasses do not belong to C. If the glasses belonged to C, that would imply that C could see and that the others could not. Since C can see in this case, he cannot be the truth teller. However, since neither A nor B can see, neither of them can truthfully say whether the first letter is E or not. This means that A could not truthfully know to say that the first letter was E, and B could not truthfully know to say that A was lying.

Following from that, this means that:

C cannot be the liar, and the glasses cannot be theirs. They are either the truth teller or the Joker, which means that one of A or B must be the liar, and the other can be either the Joker or the truth teller. We are however given the information that the truth teller HAS to be able to see to give the truth (for whatever reason) meaning he is also not the truth teller, leaving him to be the Joker.

Now we are left with an interesting observation

Since either A or B is the truth teller, the truth teller must own the glasses. If A did not own the glasses and was the truth teller he would be unable to confidently say what the first letter was. If B did not own the glasses and was the truth teller, he could not confidently say that A was lying.

Since the examiner surely knows the first letter, this means:

If the first letter is an E, then A must be the truth teller, as their statement is true, and the glasses must belong to A. If the first letter is not E, then B must be the truth teller, as their statement is true, and the glasses must belong to B.

In either case:

The examiner can confidently determine who owns the glasses.

• Some good thinking! However as your answer is based on C cannot be the liar, this will be incorrect. And this actually brings you closer to the correct answer
– Alex
Commented Apr 2, 2018 at 20:22
• If I think I have the correct answer now, should I edit this one or post another one? I don't know which is preferred Commented Apr 3, 2018 at 19:27
• you could edit this answer by adding the new one on the top and rename the current "The answer is" to "Previous attempts or first attempts / etc"
– Alex
Commented Apr 4, 2018 at 14:55

Scenario 1: Glasses belong to A

If A is telling the truth then B is lying and C is telling the truth
If A is lying, then B does not know and C is telling the truth

Scenario 2: Glasses belong to B

If B is telling the truth, then A is lying and C is telling the truth
If B is lying, then A does not know and C is telling the truth

Scenario 3: Glasses belong to C

C is lying, either A or B are telling the truth
However, neither A nor B can be the truth teller since neither of them can see, so none are able to tell the truth and the truth teller is not going to say they can see something that they cannot

Therefore:

C must be telling the truth and the glasses do not belong to C. C cannot be the liar

So:

Either A or B own the glasses and either A or B is the liar

If A owns the glasses, then B cannot know if A is lying or not (A could be the joker) so cannot say that A is lying (if B is the liar or not, so inconsistent)
If A owns the glasses and is telling the truth, then B is lying (but again, B cannot be sure, because A might be the joker, so this is also inconsistent)
If A does not own the glasses, then A is lying (they cannot see the first letter), the glasses must belong to B, who may or may not be lying (A's guess might have been right)

Basically, if A is a Joker (and A might be), then B must be the liar but cannot consistently Lie if B does not own the glasses

The only consistent answer would seem to be:

A is lying, the glasses belong to B (who may or may not be lying) and C is telling the truth

Conclusion:

The examiner can conclude that the glasses belong to B, so fails them all, but cannot be sure who the liar, truth teller or joker are (without knowing what the first letter is).

• you're the closest so far! I'm this close to accept your answer. You got the Scenario figured out - but the conclusion is indeed incorrect. Here's a bonus for your almost-correct attempt - in the scenario where C cannot be liar, does the examiner actually able to tell whose glasses belongs to? Does A or B be the liar matters?
– Alex
Commented Apr 9, 2018 at 19:34
• @Alex It might depend upon what would be a 'lie' in these circumstances. For example, if A does not know what the first letter is, is stating that it is an 'E' a lie, since A is stating that he knows what the first letter is? Commented Apr 10, 2018 at 6:18
• yes the assumption you have is correct. However what comes after this assumption is not
– Alex
Commented Apr 10, 2018 at 14:30
• @Alex - I don't think I follow you, however it does not matter whether A or B is the liar, and I don't think I said that it did. Commented Apr 10, 2018 at 15:30
• In fact, my solution does not actually resolve who the truth teller, the liar or the joker are, which is interesting. Commented Apr 11, 2018 at 8:31

Assumptions:

1) The three examinees don't know each other (so initially don't know anything about the other's predilection to tell truth or lie).

2) The three also don't know that the glasses only help the correct owner to see properly (so have no idea how well the others can see).

3) A's statement should be read as "I clearly perceive that the first letter is an E!" - i.e. A claims both that they can see the letter and that it is an E. This is to avoid a situation such as a joker mistakenly believing they can see an E (when it is really an F for example), and attempting to tell the truth.

4) The truth tellers and liars are 'classical' in the sense that:

• if T knows X=Y, then T can say X=Y
• if L knows X=Y, then L can say X<>Y
• if J knows X=Y, then J can say X=Y or X<>Y
• if T,L,or J doesn't know if X=Y, then T,L, and J cannot say either X=Y or X<>Y

I think the solution hinges on B's statement.

"A is lying!" can only be said by someone who can see properly. If B can see, then their statement is conceivably true: For example B sees that the letter is not an E, so knows A must be lying. Alternately, B's statement can be conceivably false: For example B sees that the letter is an E (assumes A is telling the truth) and lies about it. If B couldn't see then B can't be trying to tell the truth (how would B know if A lied?), and B can't be attempting to lie (because for all B knows, A could already be lying so B would inadvertently tell the truth).

So now all we need is to do is confirm if there is at least one ordering of truth teller, liar, and joker that would be consistent with the statements made. (Note that as we don't know if the letter is an E or not, we have to find a solution that is consistent in either case).

If the first letter is an E:

LJT is a consistent ordering. A doesn't know what the letter is and lies that he/she perceives an E (accidentally getting it correct). B, who can see that it is an E, mistakenly believes A has made a true statement and lies about it. C, who cant see, truthfully admits the fact. So the examiner confirms B owns the glasses.

If the first letter is not an E:

LTJ is a consistent solution. A doesn't know what the letter is and lies that he/she perceives an E (accidentally getting it wrong). B sees that the letter is not an E and tells the truth about A. C decides to be honest and admits he/she can't see. So, once again, the examiner confirms B owns the glasses.

So in either case, the examiner can confirm B owns the glasses and can fail all the examinees for cheating.

• detailed answer! For point 3) I would say the origin "It's an E" is correct, same goes for "one of the point saying "We do not know if the letter is an E". As for - "A is lying!" can only be said by someone who can see properly - it's not entirely true, B can not see clearly and lies about he could, and frame A is lying
– Alex
Commented Apr 10, 2018 at 14:29
• I believe that raises a technical issue about lying. If a liar doesn't know a fact fro sure, then can they 'lie' about the fact? In my opinion, a liar B can only state the lie 'A is lying' if B knows absolutely that A is telling the truth.I would argue that in 'truth and liars' scenarios, a liar is distinct from a fantasist - who I would define as someone who can make a statement for which they don't know the truth state. See the edit (assumption 4) to my answer. Commented Apr 10, 2018 at 20:45
• Thanks for bringing this up - I did not expect lying can be this technical. I would assume if a person claims he/she knows something when he/she does not, it's lying already. Or at least that's my initial assumption in the question. I can add clarification to the post if needed.
– Alex
Commented Apr 10, 2018 at 21:25
• That would all seem to be consistent with my solution and thinking. A liar has to lie and, although A might lie about knowing that the first letter is an 'E', B cannot lie or tell the truth with any certainty if they do not know whether A is lying or not. B could be the joker though, if A was the liar because the joker could simply not care about the truthfulness of A. Commented Apr 11, 2018 at 15:34

Here is my assumption of the given puzzle:

C = Joker, A = Truth Teller and B = Liar

So how i came to this conclusion:

As it is mentioned in the puzzle it self that the Person who have its own glasses is the one who can see properly so if C does not have its own glasses then he cannot see.So, this eliminates the fact that C is either a liar or a Truth Teller. Second how B is a liar is that they say particular things mentioned in the puzzle to the examiner not to them self, so i can assume that the examiner would not ask to them like who is lying instead of asking what is written on the board. So, B's statement tells me that he is lying and that left us with Only A who is telling the truth as for a person to tell truth he must be able to see.

Nature and Profession:

Here the joker is the only person that have its profession where as the other two are either a liar or a truth teller which is just a nature of human being. It doesn't define that wither the person who is joker by profession is a liar or a truth teller.

This is just my assumption of the details given in the puzzle.

And now the examination by examinor:

All of them can't pass the exam as they have already given their statement and it is clear that C cannot pass the exam as he already declared that he cant see and the examiner has already concluded that B is the liar by his statement is not relevant to the exam it to the person. So, we are left with only A who has passed the exam.

• Good try! However on the very first assumption on "this eliminates the fact that C is either a liar or a truthteller" is incorrect - If C does not have its glasses, he is telling the truth that he cannot see right?
– Alex
Commented Apr 11, 2018 at 14:57
• yes, i think so because i assumed that C is Joker(by profession).So, he have 50-50 chances of either telling the truth or lie. Commented Apr 12, 2018 at 6:14
• I see, that does covered 1 of the many scenario
– Alex
Commented Apr 12, 2018 at 14:24