Think methodically with common sense reasoning to solve the liar and truth-teller puzzle
Crossing over many a land a traveler came upon a deep jungle where two paths forked and went into the depths of the jungle.
He had heard from a wise man someday earlier that such a forest lay ahead. He told, "One path will lead to the warmth of a friendly village, but the other will lead you into the den of hungry tigers and sure death."
Now only he remembered the rest of what the wise old man said, "When you are the fork trying to choose the safe path, two men will suddenly appear ready to help you. Beware, one of them will be a habitual liar—his answer to any question must always be a lie, but the other will just be the opposite—he always will tell you the truth.”
The wise man finished, "Don't forget, you can ask only one question to any one the two to find the correct safe path. You won't have a second chance. They will understand your question, know which path is safe, know the nature of answering of each other but will answer only with YES or NO.”
Recommended time for you to find the safe path: 15 minutes.
This is a classic logic puzzle from old times. Logicians who juggle with pure logic by choice would immediately tell you the right answer in no time.But we are not logicians. We are common folks who use common sense logic and deductive reasoning in our own way.
If you are not a logician have a go. The experience will be interesting.
You would get better results if you imagine yourself as the traveler.
Systematic Solution to the Liar and Truth-teller Riddle – First stage: Key Pattern Discovery of the Nature of the Answer for Solving the Riddle
Without going deep into the riddle, you would do an initial experiment in mind. You will visualize what could be the answers from the two if you ask the single question,
Do you think the path to my left goes to the village that is safe?
If indeed the left path IS the safe one,
The liar would answer: NO, and the Truth-teller: YES.
Without knowing which one of them is telling the truth, you won't be able to decide.
Thinking more you realize that you have,
Two unknown paths, and two men eager to help, but again you don't know their nature of habitual answering.
With such combinations of unknowns, you realize, it would be impossible to form the right question by thinking in the conventional way. You are clear now that,
You must form the question NOT in a simple way.
Without wasting time on what type of question you have to ask, your attention shifts to the ANSWER itself.
This is a very natural way to solve problems—to analyze and understand all characteristics of the end result first, comparing it with given information.
Note: To us this is End State Analysis Approach, an often used natural problem solving technique packed with power.
Thinking more in this direction, you ask yourself the most important QUESTION at this point,
What must be the NATURE OF THE ANSWER from the two for me to know the safe path?
You have already experienced that the answer to a simple question from one helper will be NO and the other, YES, just the opposite.
You make a firm conclusion,
If the answers from the two are OPPOSITE, you won't find the safe path.
There is no going away from this. The conclusion is actually a fact and an inviolable truth.
Ah ha, you couldn't imagine earlier it would come to this pass. Yes, for you to be for sure about the safe path,
Whatever be the question, answers from both the helpers must exactly be the SAME in every situation.
This is a revelation to you and in problem solving terminology this is, discovery of the key pattern. You have now a precise requirement of their answers in relation to each other (or precise requirement specification).
Naturally, the answer will be YES or NO. But if it is NO, both will answer NO. Same must also be true for YES for knowing safe path.
This is the first breakthrough. You feel half the battle you have won by knowing all about the answer that can be known.
Nature of answer fully identified, it is time to shift focus of attention to the NATURE OF THE QUESTION.
Systematic Solution to the Liar and Truth-teller Riddle – Second stage: Process of Knowing Nature of the Question to ask
At this point you realize that you have to think in a NEW way to make the second breakthrough by understanding precisely the nature of the question which is the requirement specification for the question.
What can be a new form of a question! As you think more on this, the condition of “NEW form” strikes you and aided by the focus of the new form of question you reason,
Well, the usual simple form of the question has been a single question for any combination of situation. All combinations would have the same result ending in failure.
What can be changed about the question?
Property change analysis technique
This technique of exploring new ways to change a key property (in this case, the number of components in the question) of the key entity (in this case, the question itself) often produce great results for quick and innovative solution to the problem. This is called, Property change analysis.
Of course, you can very well change the number of component questions in the single combined question, especially as you remember from your experience,
Instead of a single question, two questions can easily be combined to form the single compound question.
When you were younger, your neighbor uncle once asked when you opened the door,
Is your father home? Is he well?
Your father was home and well. So you answered, "Yes".
One answer to two questions.
You further realize, if either your father was not home or not well or both, you would have simply answered, "No".
That must be the new way that would help you to know the safe path,
Instead of a single component single question, you would ask a single question with two components joined together by "and", the easiest way to join two questions.
And you know from elementary knowledge in language that is easy to form—your neighbor uncle could have asked such a single question,
Is your father home and well?
You feel you have practically cracked the puzzle open and that too by using no difficult technique or concept.
You have followed just simple common sense problem solving techniques coupled with concepts drawn out of everyday real life experiences and elementary knowledge of language.
What have you achieved till now?
You know now the nature of the answer and the question both.
The only task of actually forming such a question with such an answer is left.
Systematic Solution to the Liar and Truth-teller Riddle – Third stage: Forming the Question by using the Answer and Question Specifications
With clear idea of the nature of the answer and form of the question, a possible safe question to be asked would be,
Are you the truth teller and do you think the path to my left is the safe path?
Possible situations are,
Situation 1: The helper asked is the truth teller and the left path is the safe path: Final answer: YES.
Situation 2: The helper asked is the liar and the left path is the safe path, first answer result would be NO, so that answer to the combined question would truly be NO. Habitual liar, being what he is, cannot but reverse this NO to YES. Final answer: YES.
So if the answer to your question is YES, you know for sure that the left path is actually the safe path.
Alternatively, the other two possibilities are,
Situation 3: The helper asked is the truth-teller and the left path is NOT the safe path: Final answer: NO.
Situation 4: The helper asked is the liar and the left path is NOT the safe path, answer to both the parts are NO that would be reversed by the liar habitually to YES, opposite to the answer by the other helper. Final answer: YES. This violates the requirement specification of the answer.
Just on the brink of success you find that joining of two questions simply by "and" won't fully work.
There is a challenge yet to overcome at the third stage of using the specifications of the answer and question to form the right way to combine the two question components. You had assumed an easy way to combine the two questions without thinking much on question of how to combine.
But this is not all in vain—you would surely get important clues about how to combine by analyzing the results.
Systematic Solution to the Liar and Truth-teller Riddle - Final stage: Identifying Crucial Requirement of Combining two Questions
Really, why and where did combining two questions by "and" fail?
As you concentrate on finding answer to this question by analyzing the result of your last attempt, you realize that,
- Answers to the two component question 1 and question 2 are independent of each other, and,
- There is no need to think what would be the answer of the truth-teller—because primarily your goal is to force the liar to reverse the answer of the truth-teller twice to match his final answer.
In the Situation 2, using "and" for joining, you could indeed force the liar to reverse the answer of the truth-teller twice.
But in the Situation 4, where true answer to both component questions were NO, the liar reversed each to YES and combined also to YES. HE was happy to think that he had indeed reversed the true combined answer and did justice to his habit!
To force the liar to reverse true answer twice by two questions then,
One question must be DEPENDENT on the other.
So, the successful method of combining the two questions must ensure that,
The liar would reverse true answer to the INDEPENDENT first question and thinking that has answered it in line with his habit would face the second DEPENDENT question and reverse this answer more once by the question thinking again that he had answered in line with his nature.
What is the other method of joining two questions that would achieve this result?
Again your common experience of using the language helps you for the final breakthrough.
There must be only one real question but asked twice in the commonly used form,
What would be your answer if I ask you whether the "QUESTION" is true?
The first of the two component questions the liar must answer following rules of the language is,
whether the "QUESTION" is true.
The liar reverses the correct answer to this first component question and forms the INTERMEDIATE RESULT in line with his nature of answering. With this result then he faces the second component question,
What will be your answer if “INTERMEDIATE RESULT” is true?
Now he would have no option other than to reverse the intermediate result which according to him is correct and form the final result. Final result becomes the REVERSE OF INTERMEDIATE RESULT.
As INTERMEDIATE RESULT has itself been reversed once from true result, the final result returns back to the value of the true result again by this double reversal.
Following is the schematic of this mechanism,
This is pure logic no doubt, but with reason, method and two trials, you have learned enough about how the single compound question must be formed and asked to any of the two helpers to find the safe path.
With confidence you finally form the single question that would lead you to know the safe path,
What will be your answer if I ask you whether the path to my left is the safe path?
You ask this question to any of the two helpers. If the answer is YES, you take the left path, and if it is NO, you take the right path.
There cannot be any other possibility—you have indeed forced the liar to reverse the true answer twice to match the final answer of the truth-teller in both the situations.
Knowing that answer to the question from both helpers will be same, you asked the question to any one of the two helpers.
You might have discovered the most crucial PRIMARY REQUIREMENT OF ACTION—the liar must be forced to reverse the true result twice—without going into the second trial at all.
This discovery is not difficult to make if you realize early that solving the riddle will vitally depend on finding the right answer to the question,
How to make answer of the liar same as the truth-teller in any situation!
The problem solving techniques, concepts and common knowledge used
- End state analysis: Analyzing the desired result or last action first. Objective is to gain more knowledge about the last action for achieving the desired result.
- Refining requirement specification in steps: Knowing precise requirements of the answer first and the question second simplified the steps to the solution greatly.
- Question, analysis and answer or QAA technique: Simplifying the problem stage by stage by asking a series of relevant questions and analyzing each for getting its answer.
- Property change analysis technique: Exploring how many ways the key property of a key entity can be changed and promise of each change, often proves to be crucial in solving a complex problem.
- Elementary knowledge in language: Basic domain concepts: for combining two component questions to a single compound question but in different ways.
- Well-formed trials or experiments: to learn more of the problem.
- Step by step deductive reasoning: by using all of the above and discovering key patterns of information for solving the problem with complete confidence.
We have used this puzzle for a different purpose earlier but here the focus is quite different.
Our focus all through the above solution process has been to think as the traveler in a simple way and find the safe path using systematic reasoning and problem solving techniques drawn out of common knowledge and experience step by step.
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