A spy who can lie or tell the truth arrives on the Island of

A spy (who can lie or tell the truth) arrives on the Island of Knights and Knaves. You encounter three people, A, B, and C, and you know that one is a knight, one a knave, and one a spy. Each of these three knows the type of the other two.

A says: “I am the knight.”

B says: “A is not the knave.”

C says: “B is not the knave.”

Using a proof by cases, show that A must be the knave, B must be the spy, and C must be the knight.

Please show step by step, thankyou!

Solution

Dear Student

Thank you for using Chegg !!

We shall answer this question with the assumption that Knight will always speak truth, Knave will never speak truth and Spy may or may not speak truth

So the question can be divided into 2 major possibilities (which will be sub-divided into further cases)

Case 1: Spy will speak truth

Case 2:- Spy will not speak truth

Now let us first consider Case1(where only 1 of 3 people i.e. only knave will lie)

Case 1A: Let us consider that A is speaking truth that A is a knight

But then by logical arguement, B is also speaking truth that A is not knive.

And to extend further, since B is telling truth => B cannot be a knave (Remember knave always lies)

Implying that all three of them are speaking truth which is not possible.

Hence case discarded

Case 1B:- Now consider that A is not telling truth

A: A is not a knive, => A is a knave (only knave is not telling truth)

But then since B is also not telling truth by statement that A is not knave.

Implies both A & B are not telling truth.

Case discarded

Case 2: 2 out of 3 people (knight and spy) are not telling truth.

Case 2A:- A is telling truth

A:- A is a knight (Truth)

B:- A is not a knave (Truth)

=> 2 people are telling truth. Case Discarded .

Case 2B:- A is not telling truth & B is telling truth

A:- A is a knight (False)

B: - A is not a knave (Truth) => A has to be a spy and B has to be a knive since only knives are speaking truths here.

Case 2C:- Both A & B are not telling truth

A:- A is a knight (False, meaning A can be knave or spy)

B:- A is not a knave (false, So now A can only be a knave, and B can only be a spy)

Since only knaves and spy are not telling truth

C: B is not a knave (Truth,Evidently as proved above that B is a spy and hence C has to be knight telling truth)

Hence the only legitimate solution is case 2C where A is a knave, B is a spy and C is a knight.

Hence proved