logical deduction - Knights , Knaves and Spies - Part 1

I was working my way through some Knight and Knave Puzzles in Discrete Maths by Rosen, when I came across the following question:

There are inhabitants of an island on which there are three kinds of people:

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  Knights who always tell the truth

  
  


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  Knaves who always lie

  
  


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  Spies who can either lie or tell the truth.

  
  

You encounter three people, A, B, and C.

You know one of these people is a knight, one is a knave, and one is a spy.

Each of the three people knows the type of person each of other two is.

For this situation, if possible, determine whether there is a unique solution and determine who the knave, knight, and spy is :

A says "C is the knave,” B says, “A is the knight,” and C says “I am the spy"

My Solution:

$A\Rightarrow Knight$
$B\Rightarrow Spy$
$C\Rightarrow Knave$

Doubt:

Am I correct in saying my answer will work?

Answer

Yes. Your answer is correct.

A is the Knight
B is the Spy
and C is the Knave

To get the solution, First assume, A is knight and will always tells the truth.

Then as per his statement, C is the knave and so what he said will be false. That means he is not a spy. B is the spy and his statement A is the knight is random (true here). This is the only case in which the statements didn't contradict.

Now assume, A is the Knave.

Then as per his statement "C is the knave", it's clear that C is definitely not the knave. Which doesn't contradict since A is the knave already. That means, either B or C is Knight. If B is Knight his statement "A is knight" is false and it contradicts. If C is Knight his statement "I am the spy" is wrong and it contradicts. So this combination A is Knave, B is knight/Spy, C is Knight/Spy is wrong.

Continue this assumptions for other chances of combinations.

You will understand that all other combination except the first one (A is knight, B is Spy and C is knave) is wrong since the statements contradicts.