Knights And Knaves
1
Logic Puzzles
Roland Backhouse
February 13, 2001
2
Outline
Logic puzzles have been developed to test students’ skill in logical reasoning. Two classes discussed here:
• The Island of Knights and Knaves
• Portia’s Casket
Exploitation of the associativity of equivalence simplifies the problems considerably.
3
The Island of Knights and Knaves
The island has two types of inhabitants, “knights” who always tell the truth, and “knaves” who always lie.
Suppose A is the proposition “person A is a knight” and suppose A makes a statement S. Then A is true is the same as S is true. That is,
A≡S .
Examples
If A says “I am a knight” then what we can infer from the statement is A ≡ A. But since this is always true we get no information from the statement. Similarly, it cannot be that a native says “I am a knave” because we would then conclude A ≡ ¬A which is always false.
If A says “I am the same type as B”. we infer A ≡ (A ≡ B) which simplifies to B.
4
Examples (Continued)
It is rumoured that there is gold buried on the island. You ask one of the natives, A, whether there is gold on the island. He makes the following response: “There is gold on this island equivales I am a knight.” The problem is
(a) Can it be determined whether A is a knight or a knave?
(b) Can it be determined whether there is gold on the island?
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Solution
Let G denote the proposition “There is gold on the island”. A’s statement is A ≡ G. So what we are given is:
A≡A≡G .
This simplifies to G. So we deduce that there is gold on the island but it is not possible to tell whether A is a knight or a knave.
6
Formulating Questions
If native A is asked a yes/no question Q then the response to the question is
A≡Q .
That is, the response will be yes if A is a knight and the answer is really yes, or A is a knave and the answer is really no. Otherwise the response will be no.
For example, asked the question “are you a knight” all natives
Logic Puzzles
Roland Backhouse
February 13, 2001
2
Outline
Logic puzzles have been developed to test students’ skill in logical reasoning. Two classes discussed here:
• The Island of Knights and Knaves
• Portia’s Casket
Exploitation of the associativity of equivalence simplifies the problems considerably.
3
The Island of Knights and Knaves
The island has two types of inhabitants, “knights” who always tell the truth, and “knaves” who always lie.
Suppose A is the proposition “person A is a knight” and suppose A makes a statement S. Then A is true is the same as S is true. That is,
A≡S .
Examples
If A says “I am a knight” then what we can infer from the statement is A ≡ A. But since this is always true we get no information from the statement. Similarly, it cannot be that a native says “I am a knave” because we would then conclude A ≡ ¬A which is always false.
If A says “I am the same type as B”. we infer A ≡ (A ≡ B) which simplifies to B.
4
Examples (Continued)
It is rumoured that there is gold buried on the island. You ask one of the natives, A, whether there is gold on the island. He makes the following response: “There is gold on this island equivales I am a knight.” The problem is
(a) Can it be determined whether A is a knight or a knave?
(b) Can it be determined whether there is gold on the island?
5
Solution
Let G denote the proposition “There is gold on the island”. A’s statement is A ≡ G. So what we are given is:
A≡A≡G .
This simplifies to G. So we deduce that there is gold on the island but it is not possible to tell whether A is a knight or a knave.
6
Formulating Questions
If native A is asked a yes/no question Q then the response to the question is
A≡Q .
That is, the response will be yes if A is a knight and the answer is really yes, or A is a knave and the answer is really no. Otherwise the response will be no.
For example, asked the question “are you a knight” all natives