EXERCISE: A case of confused identity

Part 1.

In the former Soviet Bloc country of “Bordueria,” someone has committed a series of grisly murders.  The culprit has been identified and apprehended—with a couple of hitches.  First, the murderer has been identified by an eyewitness, but the identified individual happens to have an identical twin brother.  Both are sitting in jail, but nobody knows which one is the real murderer.  Under Bordurian law, unless some evidence points decisively to one or the other having committed the murders, both must be set free.  Also under Bordurian law, the state prosecutor ma ask each suspect only one yes/no question to ascertain his guilt or innocence.  However, here’s another hitch: one brother is a consistent pathological liar, and the other borther tells nothing but the truth, but nobody knows whether the murderer is a liar or a truth teller.

Prosecutor Multivonic think he’s figured out what question to ask each of the twins.  To each in turn, he poses the following query.

“Did your brother commit the murders?”

Will this question lead him to the murderer?

Part 2.

In response to his question, Multivonic gets a response of “Yes” from the first brother, and “No” from the second.

What can the prosecutor conclude from these answers?

Part 3.

Due to his incompetence, Multivonic is replaced as prosecutor by his assistant Plavac.  Under Bordurian law, a suspected murderer may be asked a second yes/no question by a new prosecutor—but the new prosecutor may assume no prior knowledge of the answers to any questions asked by a former prosecutor.  In other words, Plavac must start from scratch.

If you were Plavac, what would you ask the twins?  Remember, only one question is permitted for each suspect, but the question needn’t remain the same from one suspect to another.

Hint:  Don’t forget that one brother always lies, even if it is to his disadvantage.  What if in Part 2, the witness were mistaken or both brothers implicated?  When you get to Part 3 it would help to start out by asking a question which you and the brothers know the truthful answer to.

Exercise: A familial dilemma

Antonio has invited his mother and mother-in-law to meet at his house so he and his wife can take them to dinner at a nearby restaurant.  Since he has the only car, Antonio must drive them all to the restaurant.  There are, however, some complications.  His car is a Fiat two-seater convertible, so he can only take one passenger at a time.  Also, his mother-in-law doesn’t get along very well with his mother, and his wife and his mother have a rather tense relationship as well, so he can’t leave his mother along with either his wife or his mother-in-law at any point, either at home or at the restaurant.  How can he shuttle them over to the restaurant one at a time and avoid leaving two people who dislike each other alone together?  (He cannot leave anyone at a third location either.)

Hint:  There are a couple of possible answers, but they both require this: Since Antonio gets along wit all three women, but one of them can’t get along with either of the other two, what he must do id spend most of his time with that one woman and never leave her alone with either of the others.

Case of Confused Identity

Part 1 No—at least, not if one of the twins really is the murderer.  Either both will answer “Yes,” or both will answer “No,” telling the truth and who isn’t.  There are two variables here—one brother is a liar and the other is a truth-teller, and one brother is a murderer and the other is not.  If the first is a liar and answers “Yes,” then he, not his brother, is definitely the murderer.  This means that his brother is an innocent truth-teller, in which case he will also answer “Yes” in response to the question of whether his brother is a murderer.  If the first brother is a liar and answers “No,” then the other brother is the murderer—but that means that the brother is a murdering truth-teller, and will likewise answer “No” in response to the question of whether his brother is the murderer.  Either way, both brothers will give the same answer, and it will be impossible to know who is telling the truth—and who is the murderer.

Part 2  Assuming the truth of the premise that one brother is a consistent liar and the other always tells the truth, then there are only two possibilities.  The first is that the eyewitness must be mistaken—neither brother can have committed the murder.  Here’s the logic: If the first brother is a liar and answers “Yes,” then the second brother cannot be the murderer.  Also, if the first brother is a liar, the second must be telling the truth.  But then, if the second brother answers that his brother did not commit the murders, the first brother must be innocent.  Hence, both are innocent.

Part 3 The first trick is to find out which twin is the liar.  This may be done by asking the first twin any questions to which you both already know the answer.  “Do you have a twin brother?” will do.  If the first twin answers “Yes,” then you know the second twin is a liar.  Then you can ask the second twin, “Did you commit the murders?”  If he says “Yes,” then you know (since he’s a liar, and assuming that one of the twins is indeed the murderer) that the first twin is the murderer.

Note: If, on the other hand, the first win answers “No” to the first question, you know the second twin is a truth-teller, and will truthfully answer a direct question about his guilt.

A familial dilemma

First, he drives his mother over to the restaurant.  Next, he drives back and picks up his mother-in-law, and drives her to the restaurant.  When he drops off his mother-in-law, he tells his mother to get back in the car with him and takes her back home.  When he and his mother get home, he lets his mother out, picks up his wife, drives back to the restaurant, and leaves his wife at the restaurant with his mother-in-law.  Finally, he drives back home to pick up his mother again, and returns to the restaurant with her.