Solutions to a Liar, Truth-teller Puzzle: Many Ways of Reasoning

Submitted by Atanu Chaudhuri on Sun, 31/03/2024 - 19:30

How you reason is more important than the solution: Learn how to reason different ways to reach a solution

The solutions to the liar, truth-teller and random-answerer riddle shows how to analyze a logic puzzle and find many ways of reasoning to the solution.

The Liar, Truth-teller, Random-answerer Logic Puzzle

One of Arun, Barun and Chris is a knight, one a knave and one a spy. The knight always tells the truth, the knave always lies, and the spy can either tell the truth or lie. Arun says, "Chris is a knave." Barun says, "Arun is a knight." Chris says, "I am the spy." Who is the knight, who the knave and who the spy?

How many different ways can you reason out the answer? Time for you for the answer and the many ways to reach the solution is: 10 minutes.

Don't just stop after getting the solution. Ask yourself, "Can there be any other way of reasoning to find the same answer?" The more you ask such questions, you will discover new possibilities in every new problem, and it will strengthen your inherent logic analysis skill.

Solutions to the Liar, Truth-teller, Random-answerer Logic Puzzle

First reasoning: Start analyzing Chris:

We will analyze the statement of Chris first, as he identifies himself in his statement. Chances are: we'll be able to find a self-contradiction in his statement.

He says, "I am the spy."

The spy can tell either the truth or lie. Chris doesn't contradict himself as he cannot be the knight (only telling truth). He must either be the spy (who occasionally tells the truth) or the knave (who always lies).

If he indeed is the spy, Arun must be lying by his statement, "Chris is a knave." One possibility for Arun is left in this case: he must be the knave (who always lies) and so, Barun must be the knight. But, Barun cannot be the knight as he tells Arun as the knight. This is the contradiction.

In a chain of reasoning, there can either be a contradiction or a confirmation, both with complete certainty and no ambiguity.

If there is a contradiction, at the end of a reasoning chain, the original presumption must be false. Here, Chris cannot be the spy, and must be the knave. This turns out Arun telling the truth and as Barun also confirms him to be the knight.
As both of them confirms each other's statement of telling the truth, Arun must be the knight and Barun the spy (who sometimes tells the truth).

Answer: Arun is the knight, Barun the spy and Chris the knave.

We will start analyzing a different statement now.

Second reasoning: Analyze Barun's statement first

Barun's statement has the advantage of tying up two of them as well as bringing in the knight who always tells the truth. Identifying the knight or the Knave is easier by contradiction, but identifying the random-answerer is more tricky.

Barun says, "Arun is the knight."

If it is true, he can only be the occasionally truth-telling spy (he cannot be the knight as there would then be two knights).
By elimination, Chris can must only be the knave.
With all three statements tallying with each other, answer is then: Arun the knight, Barun the spy and Chris the knave.
But what if Arun lied! We would get than either a contradiction confirming the answer as unique, or we will get a second solution. For complete reasoning analysis, we need to consider Barun lying.
If Barun lied, he can be the spy or the knave, not the knight. Also Arun must not have been the knight and also lied. By elimination (there can only be two liars), Chris could not have been the liar, he must have told the truth which is self-contradictory.
So, there is no doubt that Barun could not have lied and our answer is indeed unique.

Third reasoning: Analyze statement of Chris again, but a in a different way

Chris says, "I am the spy."

He could not have been a truth-teller (it would contradict himself). He could not have been the knight always telling the truth. He must have lied.
That leaves potentially two truth-tellers (knight always, spy occasionally). Arun as the knight confirms his own statement of Chris a liar. It also confirms truth of Barun's statement and identifies himself as the spy.
Answer is same.
Logic analysis in this case focuses on number of truth-tellers and liars.

Fourth reasoning: Analyze statement of Arun first

We leave it for you.

Different way of logic analysis gives a good feel about the quick way and the longer way to reach the solution. The exercise also makes the grip on logic analysis stronger. This in general is the strength of the many ways technique of problem solving. It makes you a better problem-solver.