Bacterial and fungal abundance

The next island visited by Abercrombie was more baffling yet! It com-pounded the difficulties of a knight/knave island with those of the island of the sane and mad.

On this island, each inhabitant was either a knight or a knave; knights were truthful and knaves were liars. But half of all the inhabitants were mad and had only false beliefs, and the other half were sane and had only correct beliefs. Thus each inhabitant was one of the following four types:

(1) Sane knight (2) Mad knight (3) Sane knave (4) Mad knave

We note the following facts:

Fact 1. Anything a sane knight says is true.

Fact 2. Anything a mad knight says is false. (He tries to make true state-ments, but cannot.)

Fact 3. Anything a sane knave says is false.

Fact 4. Anything a mad knave says is true. (He tries to deceive you, but is unable to do so.)

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32 I. Be Wise, Generalize!

For example, suppose you ask an inhabitant whether two plus two is four. A sane knight knows that this is true and honestly says yes. A mad knight believes it isn’t, so, true to his belief, he says no. A sane knave knows that two plus two is four, and then lies and says it isn’t. A mad knave believes that two plus two doesn’t equal four, and then lies and says it does! Thus a mad knight answers no, but a mad knave answers yes.

Problem 5.1. Suppose you meet a native of this island and want to know whether he is sane or mad. What single yes/no question could determine this?

Problem 5.2. Suppose, instead, you wanted to find out whether he is a knight or a knave?

Problem 5.3. What yes/no question could you ask that would ensure that he will answer yes?

Problem 5.4. There is a Nelson Goodman principle for this island; one can find out any information one wants with just one yes/no question.

For example, suppose you wanted to know whether there is gold on this island. What single yes/no question would you ask?

Problem 5.5. When Abercrombie arrived on this island, he met a native named Hal who made a statement from which Abercrombie could de-duce that he must be a sane knight. What statement would work?

Problem 5.6. Abercrombie and Hal became fast friends. They often went on walks together, and Hal was sometimes quite useful in helping Aber-crombie in his investigations. On one occasion, they spied two inhabi-tants walking toward them.

“I know them!” said Hal. “They are Auk and Bog. I know that one of them is sane and the other is mad, but I don’t remember which one is which.”

When the two came up to them, Abercrombie asked them to tell him something about themselves. Here is what they said:

Auk: Both of us are knaves.

Bog: That is not true!

Which of the two is mad?

Problem 5.7. On another occasion, they came across two other natives named Ceg and Fek. Hal told Abercrombie that he remembered that one was sane and one was mad, but wasn’t sure which was which. He also

5. The Difficulties Double! 33

remembered that one was a knight and the other a knave, but again was not sure which was which.

“As a matter of fact,” said Abercrombie, “I came across these same two natives a couple of days ago and Ceg said that Fek is mad and Fek said that Ceg was a knave.”

“Ah, that settles it!” said Hal.

Hal was right. What type is each?

Problem 5.8. On another occasion, Abercrombie and Hal came across two natives named Bek and Drog, who made the following statements:

Bek: Drog is mad.

Drog: Bek is sane.

Bek: Drog is a knight.

Drog: Bek is a knave.

What type is each?

Problem 5.9 (A Metapuzzle). Several days later, after Abercrombie learned more about some of the natives, he and Hal were walking along one late afternoon and spied a native mumbling something to himself.

“I know something about him,” said Abercrombie. “I know whether he is a knight or a knave, but I don’t know whether he is sane or mad.”

“That’s interesting!” said Hal. “I, on the other hand, happen to know whether he is sane or mad, but I don’t know whether he is a knight or a knave.”

When the two got closer, they heard what the native was mumbling, which was “I am not a sane knave.” The two both thought for a while, but Abercrombie still didn’t have enough information to determine whether the native was sane or mad, nor did Hal have enough information to determine whether the native was a knight or a knave.

At this point, you have enough information to determine the type of the native. Was he sane or mad, and was he a knight or a knave?

Solutions

5.1. A particularly simple question is “Are you a knight?” A sane knight will correctly say yes; a sane knave will falsely say yes; a mad knight will incorrectly say no; and a mad knave will correctly say no. Thus, a sane inhabitant will say yes and a mad inhabitant will say no.

5.2. The question to ask is “Are you sane?” By a case analysis similar to that of Problem 5.1, you will see that a knight (whether sane or mad) will say yes and a knave will say no.

34 I. Be Wise, Generalize!

Again, we have this nice symmetry: To find out if he is sane, you ask whether he is a knight, and to find out if he is a knight, you ask whether he is sane.

5.3. Call a native reliable if he makes true statements and answers ques-tions correctly and unreliable otherwise. Reliable natives are sane knights and mad knaves; unreliable natives are mad knights and sane knaves.

A question that guarantees the answer yes is “Are you reliable?” or

“Are you either a sane knight or a mad knave?” If he is reliable, he will answer correctly and say yes. If he is unreliable, he will answer incorrectly and say yes. In either case he will say yes.

5.4. A question that works is “Are you the type who could claim that you believe there is gold on this island?” Another is “Do you believe that you are the type who could claim that there is gold on this island?”

But a much neater and simpler one is “Are you reliable if and only if there is gold on this island?”

Actually, in the next chapter, we will present an extremely general Nelson Goodman-type principle that works simultaneously for all the islands considered up to now (even the one where natives answer by flashing red or black cards) as well as the more bizarre island of the next chapter, and we will prove that it always works.

5.5. A statement that works is: “I am not a mad knave.” A sane knight could (correctly) say that; a mad knight could not (correctly) say it;

and a mad knave wouldn’t (correctly) say it. Thus, only a sane knight could say it.

5.6. We are given that one and only one of the two is sane. Suppose Auk is sane. Then he couldn’t be a knight, for then his statement would be true, which would mean that both are knaves, which is impossible if he is a knight. Therefore (assuming Auk is sane), he must be a knave. Since he is a sane knave, his statement is false, so it is not really true that both are knaves, and hence Bog must be a knight.

Also Bog is mad (since Auk is sane), so Bog is a mad knight, hence his statement is false, which would mean Auk’s statement is true, which it isn’t, since they are not both knaves. Thus, the assumption that Auk is sane leads to a contradiction. Therefore, Auk is mad.

5.7. Step 1. Suppose Ceg is the knave. Then Fek is the knight and also has made a true statement, hence Fek is a sane knight, and, therefore, Ceg is a mad knave. But then, a mad knave wouldn’t have made the false statement that Fek is a knave! Thus, it is contradictory to

5. The Difficulties Double! 35

assume that Ceg is the knave, so Ceg is the knight and Fek is the knave.

Step 2. Fek’s statement was false (since Ceg is actually a knight), and since Fek is a knave, Fek must be a sane knave. Therefore, Ceg must be a mad knight.

Thus, Ceg is a mad knight and Fek is a sane knave. (What a pair!) 5.8. Bek’s statements are either both true or both false. If both true, then

Drog is a mad knight; if both false, Drog is a sane knave. In both cases, Drog makes wrong statements. Since both of Drog’s state-ments are wrong, Bek is a mad knight. Then Bek’s statestate-ments are also both false, and so Drog is a sane knave.

5.9. Step 1. All that follows from the native’s statement is that he is not a mad knight, because a sane knight could correctly say that he is not a sane knave, and a mad knave could correctly say that, and a sane knave could falsely say that he is not a sane knave, but a mad knight could not make the correct statement that he is not a sane knave.

Step 2. Abercrombie had already known whether the native was a knight or a knave. Had he known that the native was a knight, he would then have learned from the native’s statement that he wasn’t a mad knight, and hence that he must be a sane knight. Therefore, he would have had enough information to know that he was sane.

But he didn’t have enough information, so it must be that he had previously known that the native was a knave, and hence got no additional information from knowing that he was not a mad knight.

Thus the native must be a knave.

Hal, on the other hand, had already known whether the native was sane or mad, but not whether he was a knight or knave. Had he previously known that the native was mad, then from his later knowledge that the native was not a mad knight, he would have had enough information to know that the native was a knave. But since he didn’t have enough information, then it must be that he had pre-viously known that the native was sane, and already knew that he couldn’t be a mad knight.

Thus, Abercrombie had previously known that the native was a knave, and Hal had previously known that he was sane. Conse-quently, the native was a sane knave.

Chapter 6

-A Unification

Oh, No!

When Abercrombie left the last island, he visited another one which was by far the most bizarre of all! It combined all the difficulties of all the islands previously visited. This island had the following features:

(1) Every inhabitant was classified as a knight or a knave.

(2) Male knights were truthful and male knaves were liars, but female knights lied and female knaves were truthful.

(3) Half of the inhabitants were mad and had only false beliefs, whereas the other half were sane and had only correct beliefs.

(4) When you asked a native a yes/no question, instead of answering yes or no, he or she would show you either a red card or a black card, one of which signified yes and the other, no.

(5) But different inhabitants might have meant different things by the two colors: Some of them would show a red card to signify yes and a black card to signify no, whereas some others would do the opposite!

Problem 6.1. Is there a Nelson Goodman-type principle for this crazy island? That is, can one find out any information one wants by asking just one yes/no question? For example, suppose you visit the island and want to know if there is gold on it. You meet a masked native and don’t know the sex of the native, nor whether he or she is a knight or a knave, nor whether mad or sane, nor what the colors red and black

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38 I. Be Wise, Generalize!

signify to him or her. Is there a single yes/no question that you could ask to determine whether there is gold on the island, or is no such question possible?

Problem 6.2. A related problem is this: Is there a yes/no question that will ensure that the native addressed will respond by flashing a red card?