Solution to Knights and Knaves

Some puzzles from the book "What is the name of this book" by Raymond D Smullyan

There is a wonderful book by Raymond D. Smullyan that everyone who likes puzzles should read.

Its name is: What Is the Name of This Book?: The Riddle of Dracula and Other Logical Puzzles.

Here are some puzzles from this book. All the puzzles below are about the “Island of Knights and Knaves.” Knights always tell the truth and Knaves always lie.

Puzzle 1: A stranger from out of town meets three inhabitants A, B, C. He asks A: “Are you a Knight or a Knave?” A mumbles something the stranger could not hear so he asks B: “What did he say?” B said: “He said he was a Knave.” C said: “Don’t believe B, he is lying !” What are B and C (Knights or Knaves)?

Solution: Note that a Knave can never say “I am a knave” because then he would be telling the truth. So B must be a Knave because A cannot have said: “I am a Knave.”It also follows that C must be a Knight since he made a true statement. But it is not possible to decide what A is.

Puzzle 2: Now the stranger meets two people A and B

A says: “At least one of us is a Knave.” What are A and B?

Solution: If A was a Knave then he would have told the truth when he said at least one of us is a Knave. That would be a contradiction. So, A cannot be a Knave. Hence A must be a Knight. Hence, his statement “at least one of us is a Knave” must be true. So, one of A and B is a Knight and one is a Knave. Since A is a Knight, B must be a Knave.

And here is one more that I will leave you to solve on your own:

Puzzle 3: The stranger from out of town meets three other inhabitants A, B, C. He asks A: “How many Knight are among you?” A mumbles something the stranger could not hear so he asks B: “What did he say?” B said: “He said there is one Knight among us.” C said: “Don’t believe B, he is lying !” What are B and C (Knights or Knaves)?