The Twin Brothers and the Judge

Question

Based on a Raymond Smullyan question from "Lady or Tiger".

A judge must reveal which one of them committed a certain murder. He knows that one of them always lies. He is only allowed to ask three questions and the twin brothers have to answer those questions with 'yes' or 'no'. He asks the twin brothers if they committed the murder. Their answers are insufficient for the judge to know the murderer. He asks one of them if the murderer always lies. The answer is 'no'. Now the judge knows the murderer.

Who is the murderer?

bonus question:

Does murderer always lie?

Answer 1

While one brother always lies, the other brother (in the question as currently written) does not necessarily always speak the truth.

From here on, I will refer to the brother who always lies as "Knave" (as is traditional), and the brother who may sometimes lie as "Joker". (Because, he's not a "Knight")

This gives us 4 possibilities for asking both brothers if they are the killer:

Like so:
│Killer│Answer K│Answer J│J Lying?
├──────┼────────┼────────┤
│ Joker│ Yes │ →Yes← │(Truth)
│ Joker│ Yes │ → No← │ (Lie)
│ Knave│ → No← │ Yes │ (Lie)
│ Knave│ → No← │ No │(Truth)
(Arrows indicate an answer given by someone who is the Killer in that scenario)

Now, since the Judge doesn't know who the killer is yet, we know

That both answers are the same — otherwise, the killer is the person who said "No".
We also know that if both answers are "Yes", then the Joker is the killer; but if both answers are "No" then the Knave did it.
As a bonus, it tells us that the Joker told the Truth… this time.

Next, the Judge asks one person "does the killer always lie", which gives 6 possibilities

Like so:
│Killer│Previous│Answer K│Answer J│J Lying?
├──────┼────────┼────────┼────────┤
│ Joker│ Yes │ Yes │ │
│ Joker│ Yes │ │ →Yes← │ (Lie)
│ Joker│ Yes │ │ → No← │(Truth)
│ Knave│ No │ → No← │ │
│ Knave│ No │ │ Yes │(Truth)
│ Knave│ No │ │ No │ (Lie)

As you can see, the only way that an answer of "No" tells the Judge who the killer is would be

if they both answered "Yes" to the earlier question.
By contrast: if they both answered "Yes", and the third answer was also "Yes" — or if all three answers were "No" — then the person who answers is lying… but the Judge doesn't know if they always lie, or just sometimes lie.

Which means that

The Killer is the Joker — who told the truth both times — and is also the brother of whom the Judge asked the third question.
And, since the Killer is the Joker (not the Knave) this means that the Killer does not always lie.

If the Joker had lied on the third question, or the Judge had asked the Knave instead of asking the Joker, then it would have been impossible for the Judge to tell who the killer was — because the first two questions having the same answer tells you if the Knave or the Joker is the killer, but does not tell you which brother is the Knave or the Joker.

Answer 2

Generalizing the original problem like this still does not change the answer:

The judge asks the brothers some questions and they have to answer those questions with 'yes' or 'no'. Their answers are insufficient for the judge to know the murderer. Finally, he asks one of them if the murderer always lies. The answer is 'no'. Now the judge knows the murderer.

There are four scenarios (let the last responder be X):
1. X always lies; the murderer always lies. (X is the murderer)
2. X always lies; the murderer does not always lie. (X is not the murderer)
3. X does not always lie; the murderer always lies. (X is not the murderer)
4. X does not always lie; the murderer does not always lie. (X is the murderer)

Before learning the answer, the judge does not know the murderer, so he does not rule out both 1 and 4, nor does he rule out both 2 and 3. After learning the answer, he knows the murderer, so he either rules out both 1 and 4 or both 2 and 3.

Since the answer to the question only contradicts 2, the judge could only rule out 2. Therefore, X is the murderer.

Answer 3

Let A be the brother who was asked the second question and B the other brother. We have six cases:

1. A's and B's first answers were yes/yes, respectively, and A always lies. Then A is not the murderer, and the murderer always lies. But then B is the murderer, meaning B's first answer was true, a contradiction. So we can eliminate this case.

2. A's and B's first answers were yes/yes, respectively, and B always lies. Then A is the murderer, and there is no contradiction.

3. A's and B's first answers were yes/no, respectively. Then the judge would have known that B is the murderer after the first two questions, so we can eliminate this case.

4. A's and B's first answers were no/yes, respectively. Then the judge would have known that A is the murderer after the first two questions, so we can eliminate this case.

5. A's and B's first answers were no/no, respectively, and A always lies. Then A is the murderer, and the murderer always lies, so there is no contradiction.

6. A's and B's first answers were no/no, respectively, and B always lies. Then B is the murderer. As in case 2, there is no contradiction.

To sum up, cases 1, 3 and 4 can be eliminated. If A's and B's first answers were no/no, then the judge would not know whether case 5 is true and A is the murderer, or case 6 is true and B is the murderer. Therefore, A's and B's first answers were yes/yes. So case 2 is true, and A is the murderer.

Edit: This answer was written before the OP added the bonus question. To explicitly answer that question, since case 2 is true and A is the murderer, A's first answer was true, so the murderer does not always lie.

Answer 4

Call the twin who was asked two questions "A" and the other "B". Assuming that the judge knows exactly one twin is guilty:

If the judge got one "yes" answer and one "no" answer to the first two questions, then they must be both true or both false (since exactly one twin is guilty). They can't be both true (since one brother always lies), so they are both false, and he can deduce that the twin who answered "no" is the killer.

Since he couldn't deduce the killer, he must have gotten both "yes" or both "no". He can tell whether the twin who always lies is guilty, but he doesn't know which twin that is.
If he gets two "yes, yes, no", the judge knows the always-liar is innocent. A's second answer was truthful; he is not the always liar; he is guilty.
If the judge gets "no, no, no", he knows the always liar is guilty, and A lied the second time, but we can't tell whether A always lies or just sometimes. The judge can't make a decision.

The judge was able to decide the second time, so A is guilty. For the bonus, the killer doesn't always lie.

Answer 5

Assuming one of the brothers is the murderer:

There are only two possible sets of answers to the first question: Yes/Yes, or No/No. If one brother answers Yes and the other No, that implies that either both brothers are the murderer, or neither is, in opposition to the assumption.

Then, breaking down by case:

If the answer was Yes/Yes, the murderer is telling the truth. When the judge asks "Does the murderer always lie?" and receives the answer "No", the answer is true, and the judge can deduce that they asked the second question to the murderer.

If the answer was No/No, the murderer is lying. When the judge asks "Does the murderer always lie?" and receives the answer "No", the answer is false, and again, the recipient of the second question is the murderer.

Answer 6

Overview

If the judge cannot determine from the first two questions which twin is the murderer, then he can at least determine whether the murderer is the one known always to lie. If the answer to the last question is truthful then the judge can thereby discern that only one twin always lies, and which, and therefore which one is the murderer. This must be the case, because otherwise he could not determine the murderer.

Detail

The twins are distinguished by which one is asked the latter question. Designate this twin T1, and the other T2.

If the twins had answered differently to the first question, then the judge could have reasoned that at least one of those answers must be a lie because one brother always lies. In that case, the other must also be a lie, so the murderer must be the one who denied the murder. The judge could not reason that way, so we can conclude that the brothers answered the same way to the first question.

If the twins both claimed to be the murderer then the one who is the murderer was truthful, therefore it is false that the murderer always lies. But it may be that the murderer sometimes lies. Thus, if T1 had claimed, falsely, that the murderer always lies then judge would not have had a basis to distinguish which of T1 and T2 always lies, so as to identify the murderer. On the other hand, if T1 claimed that the murderer does not always lie then that truthful statement establishes that he, alone of the two, does not always lie, and thus is the murderer.

On the other hand, if the twins both denied being the murderer then in particular, the one who always lies denied it. In that case the one who always lies must be the murderer, and the other tells the truth at least some of the time. If T1 confirms that the murderer always lies then by that truthful answer we know that he is not the one that always lies, so the murderer is T2. On the other hand, if T1 denies that the murderer always lies then that untruthful answer does not provide a means to determine the murderer.

Therefore, the possible sets of answers and final determination are:
yes, yes, no: T1
no, no, yes: T2