Seville has exactly one barber, and the following facts are known about him:
Fact 4: Bernardo is a follower of Alfredo and the barber.
Fact 5: Given any male inhabitant X, if Bernardo is a follower of Alfredo and X, then the barber is a follower of X alone.
Alfredo wears only black wigs; Bernardo wears only white wigs; Benito wears only gray wigs; Roberto wears only red wigs; and Ramano wears only brown wigs.
One Easter morning, the barber was seen wearing a wig. What color was he wearing?
SOLUTIONS
1 · The existence of Arturo alone creates no paradox, nor does the existence of Roberto alone. But it is impossible that they can both exist in the same town. (They are what Ambrose Bierce would call “incompossible”; see note following solution.) Here is the reason why:
For any person X other than Arturo, Arturo shaves X if and only if X doesn’t shave Arturo. This is true for any person X other than Arturo; so, in particular, it is true if X is Roberto. Then, taking Roberto for X, we have:
1. Arturo shaves Roberto if and only if Roberto doesn’t shave Arturo. In other words, if Arturo shaves Roberto, then Roberto does not shave him back, but if Arturo doesn’t shave Roberto, then Roberto does shave Arturo. Stated in still different terms, one of them shaves the other, but the other does not shave him back.
On the other hand, we are given that for any person X, Roberto shaves X if and only if X does shave Roberto. Taking Arturo for X, we have
2. Roberto shaves Arturo if and only if Arturo does shave Roberto.
Statement 2 says that Arturo and Roberto either both shave each other or neither one shaves the other. This is the very opposite of Statement 1, which says that one of them shaves the other, but the other doesn’t shave him back. Therefore, the given conditions are impossible; Arturo and Roberto are really “incompossible.”
Note: In The Devil’s Dictionary, Ambrose Bierce gives the following definition:
INCOMPOSSIBLE, adj. Unable to exist if something else exists. Two things are incompossible when the world of being has scope enough for one of them but not enough for both—as Walt Whitman’s poetry and God’s mercy to man. Incompossibility, it will be seen, is only incompatibility let loose. Instead of such low language as “Go heel yourself; I mean to kill you on sight,” the words, “Sir, we are incompossible,” would convey an equally significant intimation and in stately courtesy are altogether superior.
2 · No, this is no paradox. It could be that Roberto shaves himself, Arturo shaves Roberto, Arturo doesn’t shave himself, and Roberto doesn’t shave Arturo. The other X’s in the town don’t really matter; indeed, Arturo and Roberto could just as well be the town’s only inhabitants.
3 · Take any day D. We are given that E is a day such that for any inhabitants X and Y, if X shaved Y on day E, then X* shaved Y on day D. Now, let X be the official barber on day E. This means X shaved X* on day E (in fact, X* was the first one to be shaved on day E). Then, taking X* for Y, if X shaved X* on day E, then X* shaved X* on day D. And X did shave X* on day E. Therefore, X* shaved X* on day D; in other words, on day D, X* shaved himself.
The upshot is that for any day D, we let day E be as given in the conditions of the problem. Then it was not the barber of day E who necessarily shaved himself on day D, but the first one shaved by the barber on day E who must have shaved himself on day D.
4 · It is the second rumor—that there are more than a thousand members—which must be correct. Indeed, there are a lot more; the only way the given conditions can hold is that there be an infinite number of members! Let us see why:
By Fact 4, we know there is a member—call him B1—who has never been shaved at all. Now, B1 has shaved at least one member, but this member could not be B1, since B1 has never been shaved, so it must be someone else, whom we will call B2. Now, B2 has shaved someone, but it couldn’t be B1, who has never been shaved, or B2, since no one shaved himself; so it must be a new person—B3. Now, B3 has also shaved someone, but it was not B1, who has never been shaved, nor B2, since B2 was shaved by B1, nor himself, B3. So it was a new person—B4. Again, B4 couldn’t have shaved B1, nor B2, who was shaved by B1, nor B3, who was shaved by B2, nor B4, himself, so it was another person, B5. Applying the same kind of argument to B5, we see that he must have shaved some person B6 different from each of B1, B2, B3, B4, B5. Then B6 shaved some new person, B7, and so on. In this way we generate an infinite sequence of distinct members, so no finite number can suffice.
5 · Since Guido has shaved Cesare, then Guido has shaved at least one member. Therefore, all the members have shaved Guido. In particular, Lorenzo has shaved Guido. Therefore, Lorenzo has shaved at least one member, so all the members have shaved Lorenzo. In particular, Petruchio has shaved Lorenzo.
What really follows from the three conditions is that every member of the club has shaved every member!
6 · Suppose Cardano’s claim is true—we get the following contradiction:
To begin with, no member of the Exclusive Club has ever shaved himself, because he has never shaved anyone—including himself—who has shaved him. Now, suppose Cardano is a member of the club. Then he has not shaved himself, as we have just proved, hence he has failed to shave at least one member of the club—namely himself. This is contrary to his claim that he has shaved every member of the club. Therefore Cardano cannot be a member of the club.
Since Cardano is not a member of the club, then he has shaved at least one person who has shaved him. Let Antonio be such a person. Then Antonio, a club member, has shaved someone—namely Cardano—who has shaved him, which no club member can do! This is clearly a contradiction, hence Cardano’s story doesn’t hold water.
7 · Step 1: First, we prove that Roberto is a follower of the barber.
Well, consider any day on which the barber wears a wig. Either Alfredo wears a wig on that day or he doesn’t. Suppose Alfredo does. Then Bernardo also wears a wig on that day, because Bernardo is a follower of Alfredo and the barber. So Benito can’t wear a wig on that day, because he is opposite to Bernardo. Then Ramano can’t wear a wig on that day, because he wears wigs only on those days when Alfredo and Benito both do, and Benito doesn’t have one on this day. Since Ramano doesn’t wear a wig on this day, then Roberto must, because Roberto is opposite to Ramano. This proves that on any day on which the barber wears a wig, if Alfredo also does, then so does Roberto.
Now, what about a day on which the barber wears a wig but Alfredo doesn’t? Well, since Alfredo doesn’t, then it certainly is not the case that Alfredo and Benito both do; hence Ramano doesn’t, by Fact 3, and therefore Roberto does, by Fact 2. So Roberto wears a wig on any day that the barber does and Alfredo doesn’t—indeed, he wears a wig on all days that Alfredo doesn’t, regardless of the barber.
This proves that on any day on which the barber wears a wig, Roberto also does, regardless of whether Alfredo does or does not wear a wig on that day. So Roberto is indeed a follower of the barber.
Step 2: We next prove that Bernardo is a follower of Alfredo and Roberto.
Consider any day on which Alfredo and Roberto both wear wigs. Benito can’t wear a wig on that day, because if he did, then Alfredo and Benito both would have them on, which by Fact 3 would mean that Ramano also does, and we would have Ramano wearing a wig on the same day as Roberto, which is contrary to Fact 2. Therefore, on any day on which Alfredo and Roberto both wear wigs, Benito doesn’t, and hence Bernardo does, according to Fact 1. This proves that Bernardo is a follower of Alfredo and Roberto.
Step 3: Now we are in a position to show that the barber is a follower of Roberto, which is the converse of what we proved in Step 1.
To do this, we use Fact 5 for the first time. Fact 5 is true for every male inhabitant X, so in particular it holds if X is Roberto. Therefore, we know that if Bernardo is a follower of Alfredo and Roberto, then the barber is a follower of Roberto alone. And Bernardo is a follower of Alfredo and Roberto, by Step 2. Therefore, the barber is a follower of R
oberto.
Step 4: Now we know that Roberto is a follower of the barber, by Step 1, and that the barber is also a follower of Roberto, by Step 3. Therefore, Roberto and the barber wear wigs on exactly the same day. But we were given that no two different people wear wigs on exactly the same days. Hence Roberto and the barber mut be the same person! Finally, since Roberto wears only red wigs, the barber can wear only red wigs. So the answer to the problem is red.
4
The Mystery of the Photograph
INTRODUCTION
For the benefit of my new readers, I had best review an old puzzle of mine.
Suppose we consider, as in Chapter 2, a pair of brothers, one of whom always tells the truth and the other of whom always lies. But now we have a new complication. The truth-teller is completely accurate in all his judgments; all true propositions he knows to be true, and all false propositions he knows to be false. On the other hand, the lying brother is totally inaccurate in his judgments; all true propositions he believes to be false and all false propositions he believes to be true. Now, in fact, whatever question you ask of either brother, you will get the same answer. For example, suppose you ask whether two plus two equals four. The accurate truth-teller will know that two plus two equals four and will honestly answer yes. The inaccurate liar will erroneously believe that two plus two doesn’t equal four, but then he will lie and claim that it does equal four, and so he will also answer yes.
The situation is reminiscent of an incident I once read about in a psychiatric journal and reported in What Is the Name of This Book? The doctors of a certain mental institution were thinking of releasing a schizophrenic patient. They decided first to give him a lie detector test. One of the two questions they asked him was: “Are you Napoleon?” He replied, “No.” The machine showed he was lying!
Getting back to my puzzle, suppose the two brothers are identical twins who are indistinguishable in appearance. You meet one of them alone and wish to find out whether he is the accurate truth-teller or the inaccurate liar. Is it possible to do this by asking any number of yes/no questions? According to one argument, it is not possible, because whatever question you ask of either brother, you will get the same answer (as I have proved). But there is another argument—which I won’t yet tell you—that it is possible. Is it really possible or isn’t it?
The answer is yes, it is possible; moreover, it can be done in only one question! All you need ask is: “Are you the accurate truth-teller?” If he is, then he will know that he is, since he is accurate, and will honestly reply yes. But if he is the inaccurate liar, then he will believe he’s the accurate truth-teller, since his beliefs are all wrong, and will lie about his beliefs and answer no. So if he answers yes, you will know he is the accurate truth-teller; if he answers no, you will know he is the inaccurate liar.
But doesn’t this raise a paradox? On the one hand, I have proved that the two brothers will give the same answer to the same question, yet I have just exhibited a question to which they would give different answers! How can this be? Was my first proof fallacious?
The answer is no, the proof was perfectly valid; both brothers will indeed respond the same way to the same question, and in fact will give the correct answer. The whole point is that the six words “Are you the accurate truth-teller?” when asked of one person constitute a different question than when they are asked of another, because they contain the variable term “you,” whose denotation depends on the person addressed! And so you are not really asking the same question, even though the sequence of words is the same.
The technical adjective for words like “you,” “I,” “this,” “that,” and “now” is indexical. Their denotations are not absolute, but depend upon their contexts. This “indexical” principle was delightfully used by Ambrose Bierce in his book The Devil’s Dictionary. After defining the word “I,” he goes on to say: “The plural of ‘I’ is said to be we, but how there can be more than one myself is doubtless clearer to the grammarians than it is to the author of this incomparable dictionary.”
THE CASE OF THE FOUR BROTHERS
Now we have four brothers, named Arthur, Bernard, Charles, and David. The four are quadruplets and are indistinguishable in appearance. Arthur is an accurate truth-teller; Bernard is an inaccurate truth-teller (he is totally deluded in all his beliefs but always states honestly what he does believe); Charles is an accurate liar (all his beliefs are correct, but he lies about every one of them); and David is an inaccurate liar (he is both deluded and dishonest; he tries to give you false information but he is unable to!).
You see, of course, that Arthur and David will both give correct answers to any question asked, whereas both Bernard and Charles will give the wrong answer to any question asked.
1 • A Simple Starter
Suppose you meet one of the four brothers one day on the street. You wish to find out his first name, and you are allowed to ask only yes/no questions. What is the smallest number of questions you need ask, and what would the questions be?
• 2 •
Arthur and Bernard are both married; the other two brothers are not. Arthur and Charles are both wealthy; the other two brothers are not.
You meet one of the four brothers one day and wish to find out whether he is married. What yes/no question would you ask? This can be done with a three-word question!
• 3 •
Suppose, instead, you wanted to find out if he is wealthy. What question would you ask?
• 4 •
I once met one of the four brothers and asked him a yes/no question. I should have realized before I asked it that the question was pointless, because I could have known in advance what the answer would be. Can you supply such a question?
• 5 •
Suppose someone makes you the following offer. You are to interview one of the four brothers and try to find out which one he is. You may ask him one question, or you may ask him two questions, but you must decide in advance which you will do. If you choose the two-question option and ascertain his identity, you will be given a prize of a hundred dollars. But if you choose the one-question option and can determine his identity, you will get a thousand dollars. However, under this option, if you fail after the first question, you are not allowed to ask a second.
From the point of view of pure mathematical probability, would you choose the one-question option or the two-question option?
• 6 •
In preparation for the two special puzzles, 8 and 9, that follow, I wish to illustrate a basic principle.
As you already know, if you ask any of the four brothers whether two plus two equals four, Arthur and David will answer yes, and Bernard and Charles will answer no. Now, suppose you instead ask: “Do you believe that two plus two equals four?” What will each brother answer?
• 7 •
Suppose you ask one of the brothers whether two plus two equals four and he answers no. Then you ask him whether he believes that two plus two equals four and he answers yes. Which of the four brothers is it?
TWO SPECIAL PUZZLES
8 • A Metapuzzle
One day a logician came across one of the four brothers and asked him, “Who are you?” The brother identified himself as either Arthur, Bernard, Charles, or David, and the logician then knew who he was.
A few minutes later, the same brother was met by a second logician, who asked him, “Who do you believe you are?” The brother answered, again either Arthur, Bernard, Charles, or David, and the second logician then knew who he was.
Who was he?
9 • The Mystery of the Photograph
If you ever visit these four brothers at their home, you will notice a photograph of one of them in the living room. If you ask each of them whether it is his photograph, three of them will answer no and one will answer yes. If you ask each one whether he believes it is his photograph, then again three will answer no and one will answer yes.
Whose photograph is it?
SOLUTIONS
&nbs
p; 1 · Two questions are enough, and these can be chosen in many ways. Here is one sequence:
You first ask the brother you meet whether two plus two equals four. If he answers yes, then you know that he answers all questions correctly and must therefore be Arthur or David. Then you simply ask him whether he is Arthur and you abide by what he answers. If he answers no to your first question, then you know that he answers all questions incorrectly and that he must be either Bernard or Charles. Then you ask him if he is Bernard and you abide by the opposite of what he answers.
2 · To find out whether he is married, you need only ask him: “Are you wealthy?” Arthur will answer yes, since he is wealthy and gives correct answers; Bernard will also answer yes, since he is not wealthy and gives wrong answers; Charles will answer no, since he is wealthy but gives wrong answers; and David will answer no, since he is not wealthy and gives correct answers. And so a married brother will answer yes and an unmarried brother will answer no.
3 · To find out if he is wealthy, you ask: “Are you married?” As the reader can check, Arthur and Charles, the wealthy brothers, will answer yes and Bernard and David will answer no.
4 · The question I stupidly asked was: “Are you either Arthur or David?” I should have known that any of the four brothers would answer yes, since if he was either Arthur or David, he would correctly answer yes; if he was Bernard or Charles, he would incorrectly answer yes.
5 · You are better off choosing the one-question option! With the two-question option, you have a certainty of winning a hundred dollars by following the procedure of the solution of Problem 1, but with the one-question option, you have a one-out-of-four chance to win a thousand dollars. Just ask the interviewee if he is Bernard. If he is Charles, he will answer yes; each of the other three will answer no, as you can check. So if you get yes for an answer, you will know he is Charles, and you can claim your prize. If you get no for an answer, you won’t know which of the other three he is, and so you cannot then collect. But a one-quarter chance at a thousand dollars is mathematically better odds than a certain win of a hundred dollars.
To Mock a Mocking Bird Page 3
