To Mock a Mocking Bird
Page 4
6 · Remember that you are asking the brother whether he believes that two plus two equals four. Arthur would obviously answer yes. Now, what about Bernard? Many readers will be tempted to say that Bernard will answer no, but this is not so! Bernard would also answer yes, and here is why:
Bernard, being inaccurate, doesn’t believe that two plus two equals four. But Bernard is inaccurate in all his beliefs—even his beliefs about his own beliefs! And so he will wrongly believe that he does believe that two plus two equals four! Then, being a truth-teller, he will honestly claim what he believes—namely that he believes that two plus two equals four.
A possibly simpler way of seeing the situation is this: Bernard doesn’t believe that two plus two equals four, and so the correct answer to the question is no. But he gives only incorrect answers to questions, so he answers yes.
Now, with Charles, it is a different story. Charles does believe that two plus two equals four, and being accurate, he believes that he believes that two plus two equals four. But he is a liar, hence he denies that he believes that two plus two equals four, and answers no.
David doesn’t believe that two plus two equals four, and he answers all questions correctly, so he will answer no.
In summary, Arthur and Bernard, the truth-tellers, answer yes and Charles and David, the liars, answer no to the question of whether they believe that two plus two is equal to four.
7 · Only Bernard and Charles can deny that two plus two equals four. Only Arthur and Bernard can claim to believe that two plus two equals four, as we have seen from the last problem. So the only brother who could answer no to the first question and yes to the second is Bernard.
8 · We leave it to the reader to verify the following four facts:
1. Arthur, Charles, and Bernard could each claim to be Arthur.
2. Only Charles could claim to be Bernard.
3. Only Bernard could claim to be Charles.
4. Bernard, Charles, and David could each claim to be David.
Therefore, if the first logician had received either the answer “Arthur” or “David,” he couldn’t have known which brother he was addressing. But he did know, so he either got the answer “Bernard” and knew that the brother was really Charles, or he got the answer “Charles” and knew that the speaker was really Bernard. So now you know that the brother was either Bernard or Charles, but you don’t know which, although the first logician did.
Now for the question of the second logician. Using principles illustrated in the last two problems, the reader should be able to verify the following four facts:
1. Arthur, Charles, and David could each claim to believe he is Arthur.
2. Bernard, Charles, and David could each claim to believe that he is Bernard.
3. Only David could claim to believe he is Charles.
4. Only Charles could claim to believe he is David.
So the only way the second logician could have been sure of the identity of the brother is that either he got the answer “Charles” and knew that the speaker was David, or he got the answer “David” and knew that the speaker was Charles. We already know that he is not David—because he is either Bernard or Charles—so he must be Charles.
9 · As the reader can verify, if the photograph is of Arthur, then three of the brothers, Arthur, Bernard, and Charles, would answer yes to the first question, and therefore the photograph is not of Arthur. If the photograph were of David, again you would get three yes answers to the first question, from Bernard, Charles, and David, so the photograph is not of David. With Bernard’s photograph, you would get three no answers, by Arthur, Bernard, and David, and with Charles’s photograph you would also get three no answers, by Arthur. Charles, and David. And so from the fact that three of the answers to the first question are no, the photograph must be of Bernard or Charles
As to the second question, if the photograph is of Bernard, you would get only one no answer, from Arthur, but with Charles’s photograph, you would get three no answers, from Arthur, Bernard, and Charles. Therefore the photograph is of Charles.
PART TWO
KNIGHTS, KNAVES,
AND THE
FOUNTAIN OF YOUTH
5
Some Unusual Knights and Knaves
My earlier puzzle books—What Is the Name of This Book?, The Lady or the Tiger?, and Alice in Puzzle-Land—are chock-full of puzzles about an island in which every inhabitant is either a knight or a knave, and knights make only true statements and knaves only false ones. These puzzles have proved popular, and I give some new ones in this chapter. First, however, we will consider five questions that will serve both as an introduction to knight-knave logic for those not familiar with it and as a brief refresher course for those who are. Answers are given following the fifth question.
Question 1: Is it possible for any inhabitant of this island to claim that he is a knave?
Question 2: Is it possible for an inhabitant of the island to claim that he and his brother are both knaves?
Question 3: Suppose an inhabitant A says about himself and his brother B: “At least one of us is a knave.” What type is A and what type is B?
Question 4: Suppose A instead says: “Exactly one of us is a knave.” What can be deduced about A and what can be deduced about B?
Question 5: Suppose A instead says: “My brother and I are the same type; we are either both knights or both knaves.” What could then be deduced about A and B? Suppose A had instead said: “My brother and I are different types.” What can then be deduced?
Answer 1: No; no inhabitant can claim to be a knave because no knight would lie and say he is a knave and no knave would truthfully admit to being a knave.
Answer 2: This question has provoked a good deal of controversy! Some claim that anyone who says that he and his brother are both knaves is certainly claiming that he is a knave, which is not possible, as we have seen in the answer to Question 1. Therefore, they conclude, no inhabitant can claim that he and his brother are both knaves.
This argument is wrong! Suppose an inhabitant A is a knave and his brother B is a knight. Then it is false that he and his brother are both knaves, hence he, as a knave, is certainly capable of making that false statement. Therefore it is possible for an inhabitant to claim that he and his brother are both knaves, but only if he is a knave and his brother is a knight.
This illustrates a curious principle about the logic of lying and truth-telling: Normally, if a truthful person claims that both of two statements are true, then he will certainly claim that each of the statements is true separately. But with a constant liar, the matter is different. Consider the following two statements: (1) My brother is a knave; (2) I am a knave. A knave could claim that (1) and (2) together are both true, provided his brother is actually a knight, but he cannot claim (1) and claim (2) separately, since he cannot claim (2). Again, a knave could say: “I am a knave and two plus two is five,” but he cannot separately claim: (1) “I am a knave”; (2) “Two plus two is five.”
Answer 3: A says that, of A and B, at least one is a knave. If A were a knave, then it would be true that at least one of A and B is a knave and we would have a knave making a true statement, which is not possible. Therefore A must be a knight. Since he is a knight, his statement is true, hence at least one really is a knave. It is then B who must be the knave. So A is a knight and B is a knave.
Answer 4: A is saying that exactly one of the persons A and B is a knave. If A is a knight, his statement is true, exactly one is a knave, and so B is a knave. If A is a knave, his statement is false, hence B must again be a knave, because if B were a knight, then it would be true that exactly one is a knave! And so regardless of whether A is a knight or a knave, B is a knave. As for A, his type cannot be determined; he could be either a knight or a knave.
Answer 5: If B were a knave, no native would claim to be the same type as B, because that would be tantamount to claiming to be a knave. Therefore B must be a knight, since A did claim to be of the same type
as B. As for A, he could be either a knight or a knave.
If A had instead said that he and B were different types, this would be equivalent to the statement “One of us is a knight and one of us is a knave,” which in turn is the same as the statement “Exactly one of us is a knave.” This is really the same as Question 4, and so the answer is that B is a knave and A is indeterminate.
Looked at another way, if B were a knight, then no inhabitant would claim to be a different type than B!
Now that the review is over, the fun will start!
THE SEARCH FOR ARTHUR YORK
1 • The First Trial
Inspector Craig of Scotland Yard—of whom you will read much in this book—was called to the Island of Knights and Knaves to help find a criminal named Arthur York. What made the process difficult was that it was not known whether Arthur York was a knight or a knave.
One suspect was arrested and brought to trial. Inspector Craig was the presiding judge. Here is a transcript of the trial:
CRAIG: What do you know about Arthur York?
DEFENDANT: Arthur York once claimed that I was a knave.
CRAIG: Are you by any chance Arthur York?
DEFENDANT: Yes.
Is the defendant Arthur York?
2 • The Second Trial
Another suspect was arrested and brought to trial. Here is a transcript of the trial:
CRAIG: The last suspect was a queer bird; he actually claimed to be Arthur York! Did you ever claim to be Arthur York?
DEFENDANT: No.
CRAIG: Did you ever claim that you are not Arthur York?
DEFENDANT: Yes.
Craig’s first guess was that the defendant was not Arthur York, but are there really sufficient grounds for acquitting him?
3 • The Third Trial
“Don’t despair,” said Craig to the chief of the island police, “we may find our man yet!”
Well, a third suspect was arrested and brought to trial. He brought with him his defense attorney, and the two made the following statements in court.
DEFENSE ATTORNEY: My client is indeed a knave, but he is not Arthur York.
DEFENDANT: My attorney always tells the truth!
Is there enough evidence either to acquit or convict the defendant?
SOME UNUSUAL KNIGHTS AND KNAVES
The puzzles of this section are quite unlike any of my past puzzles about knights and knaves.
4 • My First Adventure
Inspector Craig left the island shortly after winding up the case of Arthur York. Two days later, I came to this island looking for adventure.
On the first day I arrived, I met an inhabitant and asked him: “Are you a knight or a knave?” He angrily replied: “I refuse to tell you!” and walked away. That’s the last I ever saw or heard of him.
Was he a knight or a knave?
5 • My Second Day
On the next day I came across a native who made a certain statement. I thought for a moment and said: “You know, if you hadn’t made that statement, I could have believed it! Before you said it, I had no idea whether it was true or not, nor did I have any prior knowledge that you are a knave. But now that you have said it, I know that it must be false and that you are a knave.”
Can you supply a statement that could fulfill those two conditions? Note: The statement “Two plus two is five” won’t work; I would have already known that statement to be false before he made it.
6 • The Next Day
On the next day I came across a native who said: “My father once said that he and I are different types, one a knight and one a knave.”
Is it possible that his father really said that?
7 • The Next Day
On the next day I was in a rather frivolous mood. I passed a native and asked him: “Do you ever answer no to questions?” He answered me—that is, he said either yes or no—and I then knew for sure whether he was a knight or a knave. Which was he?
The above puzzle occurred to me as a result of a clever story told to me by the mathematician Stanislaw Ulam. Ulam referred to it as a paradox. The story is a true one and refers to a certain past president of the United States.
On television Professor Ulam saw this president address the cabinet on his first day in office. He said to them, in a supercilious tone of voice: “You men are not all yes-men, are you?” They all solemnly replied: “Noooo!”
And so it seems that a person doesn’t necessarily have to answer yes to be a yes-man! I am also reminded of a cartoon sent to me by one of my readers. It is a drawing of a tough-looking employer saying to his meek-looking employee: “I hate yes-men, Jenkins; don’t you?”
8 • The Sociologist
On the next day, I met a sociologist who was visiting the island. He gave me the following report:
“I have interviewed all the natives of this island and I have observed a curious thing: For every native X, there is at least one native Y such that Y claims that X and Y are both knaves.”
Does this report hold water?
9 • My Last Adventure
My last adventure on this island during that particular visit was a curious one. I met a native who said, “This is not the first time I have said what I am now saying.”
Was the native a knight or a knave?
SOLUTIONS
1 · If the defendant is Arthur York, we get the following contradiction. Suppose he is Arthur York. Then he is a knight, since he claimed to be Arthur York. That would mean that his first answer to Craig was also true, which means that he, Arthur York, once claimed that he was a knave. But that is impossible! Therefore the defendant is not Arthur York, although he is, of course, a knave.
2 · The defendant is either a knight or a knave. Suppose he is a knight. Then his answers were both truthful; in particular, his second answer was truthful, so he did once claim that he is not Arthur York. His claim was true, since he is a knight; thus he is not Arthur York. This proves that if he is a knight, then he is not Arthur York.
Suppose he is a knave. Then his answers were both lies; in particular, his first answer was a lie, which means that he did once claim to be Arthur York. But since he is a knave, he lied when he claimed to be Arthur York, hence he is not Arthur York. And so we have proved that if he is a knave, then he is not Arthur York.
We now see that regardless of whether he is a knight or a knave, he cannot be Arthur York. And so he was acquitted. Incidentally, it cannot be determined whether he is a knight or a knave.
3 · It was very stupid for the defendant to say what he did! Of all the false statements he could have made, he chose just about the most incriminating one possible. Here is why the defendant must be Arthur York:
Suppose the defense attorney is a knight. Then his statement is true, which implies that the defendant is a knave. Hence the defendant’s statement is false, which means that the defense attorney is a knave. So if the defense attorney is a knight, he is also a knave, which is impossible. Therefore the defense attorney can’t be a knight; he must be a knave. It then follows that the defendant is also a knave, since he falsely claimed that his attorney always tells the truth. And so we now know that both the attorney and the defendant are knaves.
Now, if the defendant were not Arthur York, then it would be true that the defendant is a knave but not Arthur York, hence the attorney would have made a true statement. But the attorney is a knave and can’t make a true statement! Therefore the defendant must be Arthur York.
4 · He said that he refused to tell me, and by Gad he did refuse! So he told the truth; hence he was a knight.
5 · There are many possible statements that would work. What actually happened was this:
Before he spoke, I had no idea whether he was a knight or a knave, nor did I know whether he was wealthy or not. But then he said: “I am a wealthy knave.” A knight could never say he was a wealthy knave, hence I realized that he must be a knave, but not a wealthy one.
A clever alternative solution suggested to me by a brigh
t high school student is: “I am mute.”
6 · If he hadn’t said that his father once said that they were different types, then it would have been possible for his father to have said it. But suppose the father had really said that. Then the father must be a knight and the son must be a knave—see Question 5 and its answer, at the beginning of this chapter—and therefore the son would never have truthfully said that his father said that.
Incidentally, the statement “My father once said that he and I are of different types” provides yet another solution to Problem 5.
7 · If he had answered yes, I would have concluded that he was probably a knight. But he answered no, and so I knew for sure that he was a knave, because he just answered no, thus falsely denying that he ever answered no!
8 · If the report is true, we get the following contradiction. For every X there is some Y who claims that X and Y are both knaves. Now, the only way that Y can claim that X and Y are both knaves is that Y is a knave and X is a knight. Therefore every inhabitant X of the island must be a knight. Yet for every inhabitant X there is at least one inhabitant Y who is a knave, since he claims that X and Y are both knaves. So there is at least one knave Y on the island. This contradicts the already proved fact that all the inhabitants are knights.