## Outline Mathematics

Logic Problems

# Knights and Knaves 2

Here's a problem to tackle:

On an island, the populace is of two kinds: knights and knaves. Knights always tell the truth, knaves always lie.

An islander - call him A - made a statement about himself and a friend, call him B: "Either I am a knave or B is a knight."

What are A and B?

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Copyright © 1996-2017 Alexander Bogomolny

### Solution

On an island, the populace is of two kinds: knights and knaves. Knights always tell the truth, knaves always lie.

An islander - call him A - made a statement about himself and a friend, call him B: "Either I am a knave or B is a knight."

What are A and B?

Recollect that disjunction (this or that) is false if and only if both this and that are false,both this and that are false,at least one of this or that is false. It is true otherwise, in particular if either this or that is true,false,true. Assume A is a knave. Then what he said happens to be true,false,true. But knaves always lie,lie,tell truth. So our assumption that A is a knave,knave,knight is wrong,wrong,correct. A must be a knight,knave,knight and what he said is true,false,true. Since the first part of his statement 'I am a knave' is false,false,true, in order for the whole disjunction 'this or that' to be true, the second part 'B is a knight' ought to be true. Therefore, A is a knight,knave,knight and B is also a knight,knave,knight.

### References

- R. Smullyan,
*What Is the Name of This Book?*, Simon & Schuster, 1978

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Copyright © 1996-2017 Alexander Bogomolny

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