# Magicians, Knights and Knaves

My solution for this puzzle did not well match with the book's solution. I thought it would be good to check my argument's reasoning, though you can also propose your own answer.

There is an island made out of only knights (that tell only truth) and knaves (that tell only lies). I encounter three inhabitants of it: A,B,C. I know that one of them is a magician (There is no characterization of a magician). Three of them made the following statements:

A: B is not both a magician and a knave.
B: Either A is a knave or I am not a magician
C: Magician is a knave

Who is the magician?

I gave an argument like this (it might have mistakes):

C can not be a magician because:
- if C is a knight, he must be a knave because of the truthfulness of his own statement.
- if C is a knave, then his statement is indeed true given the situation and thus, he is a knight.
This argument is enough to justify that C is not a magician.

I might have mistakes in this part of the argument (Please inform if it is wrong)

Now, A and B can not lie simultaneously because then, B must be both a magician and knave. But B, since A is a knave, is in fact telling the truth. Thus, B is a knight. This contradicts the hypothesis. Further, if A and C are simultaneously true, then, B must be a magician as A and C can not be magicians(as they are knights). But then due to C, B is a knave. But that contradicts the truthfulness of A. If B and C are true simultaneously, then A must be a magician and a knave. But A is knight since A made a correct statement about the situation of B. From this, we conclude that C must be false. Thus, The magician must be a knight. If A is a knave, then B is a knave and a magician. But A and B can not be liars simultaneously. Thus, A is a knight. If B is a liar, then he is making a true statement because he can not be a magician because of being a liar. Thus, B is true. Thus, A must be a magician.

This puzzle is from Raymond Smullyan's books The Gödelian Puzzle Book: Puzzles, Paradoxes and Proofs.
The book's solution is:

From C’s statement it follows that C cannot be the magician, because if he is a knight then the magician is really a knave and hence cannot be C. On the other hand if C is a knave, then contrary to his statement, the magician is not a knave but a knight, hence cannot be C who is a knave. Thus in either case, C is not the magician.
Next we will see that A must be a knight. Well, suppose he were a knave. Then his statement is false, which means that B must be both a knave and a magician.
Since A is a knave (under our assumption) then it is true that either A is a knave or (anything else!). Thus it is true that either A is a knave or B is not the magician, but this is just what B said, and thus the knave B made a true statement which is not possible! Thus the assumption that A is a knave leads to an impossibility, hence A cannot be a knave. Thus A is a knight. Hence his statement is true, which means that B is not both a knave and a magician.
We now know the following:
(1) C is not the magician.
(2) A is a knight.
(3) B is not both a knave and a magician.
Next we will see that B cannot be the magician, for suppose he were. Then it is false that he is not the magician, and it is false that A is a knave (by (2)), hence both alternatives of B’s statement are false. Hence B’s statement must be false, which makes B a knave. Hence B is then both a knave and a magician, which is contrary to (3)! Thus it cannot be that B is the magician. Also C is not the magician, as we have seen. Thus it must be A who is the magician.
Also, since B is not the magician, what he said is true, hence B is a knight. As for C, what he said cannot be true, since the magician is really a knight (A), not a knave. Hence C is a knave.
In summary, A and B are both knights, C is a knave and the magician is A.

• Your solution seems to be the same to the correct solution and to the solution of the book: A&B are knights, C is knave, A is magician. Which part do you think is different? Aug 20, 2021 at 7:26
• The way I got the solution does not match. The first part of both our arguments match. But then I proceed to prove that A and B can not simultaneously lie,B and C cannot simultaneously tell the truth and A and C cannot simultaneously tell the truth. Aug 20, 2021 at 7:32
• But the book tried to consider singular cases and then get the final answer. Aug 20, 2021 at 7:34
• I wanted to validate the argument's correctness not the answer's.Sorry for confusion. Aug 20, 2021 at 7:36

I got a bit lost in your explanation so I'm going to give my own path of logic.

As you correctly pointed out,

C is not the magician, because if C is a knight, the magician is a knave, and if C is a knave, the magician is a knight.

From there, we have cases.

Case 1: A is a knave

A's statement can be reversed to show B is both magician and knave. B is a knave, and one clause of the OR statement (A is a knave) is true, so B is not lying. This case fails.

Case 2: A is a knight, B is a knave

B is not both a magician and a knave, so he cannot be the magician, and therefore A is the magician. However this breaks, as the B's statement "I am not the magician" is true, making his OR statement true, despite B being a knave. This case fails.

Case 3: A & B are both knights

A's statement works because B is not a knave. Looking at B's statement, we know that A has to be magician because A is not a knave, so B cannot be magician in order for his statement to be true.