## Outline Mathematics

Logic Problems

# Knights and Knaves 1

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: "At least one of us is a knave."

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: "At least one of us is a knave."

What are A and B?

Assume A is a knave. Then what he said - at least one of them is a knave,knave,knight - is false,true,false. But knaves always lie,lie,tell the truth. Thus, as a knave, A could not have uttered a correct sentence. Our assumption is therefore wrong and A is a knight,knave,knight. As such, he always speaks the truth,lies,the truth. Therefore, one of them is a knave,knave,knight. Since A,A,B is a knight, B,A,B is bound to be a knave.

### References

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

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