Solve liar and truth-teller riddle second version by common sense reasoning and method
In Liar, Truth-teller, Random-Answerer Riddle, a traveler can ask two questions to decide which is the safe path out of two. What to ask?
Story of the Liar, Truth-teller, Random-answerer Riddle
Many a month have passed since the traveler narrowly escaped death by asking the right question to one of the liar and truth-teller who were ready to help.
Now again the traveler came upon a deep jungle where two paths forked and went into the depths of the jungle.
He knew of this tough challenge that he is facing from the wise man already. The wise man had said, "You'll again face only two paths going into a deep jungle second time. One path will lead to the warmth of a friendly village, but the other will lead you to a village of cannibals who will eat you."
The wise old man continued, "This time when you are at the fork trying to choose the safe path, three 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, a second one will just be the opposite—he always will tell you the truth, but the third one will be the most unpredictable. He will answer to your question absolutely randomly.”
Then wise man finished, "Don't forget, you can ask only two questions to any two of the three one by one to find the correct safe path. They will understand your question, know which path is safe, know the nature of answering of each other as well but will answer only with YES or NO.”
Recommended time for you to find the safe path: 45 minutes.
This is a more difficult second version of the classic liar and truth-teller logic puzzle from old times. Logicians who juggle with pure logic by choice won't find it difficult to decide the right questions to ask. But we are not logicians. We are common folks who use common sense logic and deductive reasoning in our own way.
Add to that our previous experience of solving a simpler version of the puzzle. That will surely help, though not necessarily needed.
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, truth-teller and random-answerer riddle – Stage 1: Decision to analyze the final situation
Without bothering about any other details of the riddle, you'll totally focus on the two-part end point when you are asking the second question,
Part 1: What must be the knowledge gained after the first question regarding the nature of the three helpers when you ask that second question?
Part 2: What can be the second and last question to ask?
This is a natural decision that any one should make and it follows the powerful but natural problem solving technique of End state analysis approach.
Truly, it should be much easier to analyze the second question situation to determine whether and how the traveler can decide the right path. The situation is simpler as you would have only one question left, as well as you have already covered some grounds in knowing the nature of the three helpers by your first question.
Solution to liar, truth-teller, random-answerer riddle - Stage 2: Specifying necessary requirements when you ask second question
This time the chief problem maker is the random-answerer as you can't take any decision based on his answer.
The obvious first conclusion is a certainty,
Conclusion 1: When you ask the second and FINAL question, YOU HAVE TO ASK IT TO ANY ONE OF THE TWO OF THE HELPERS NOT THREE. And none of these two can be the randon-answerer.
Naturally you have to eliminate one of the three helpers by your fitst question so that when you ask the second question, you ask it to one of two helpers not three. This is the only way you can solve the challenge. And, more importantly,
If the random-answerer by chance is one of two whom you ask the second question, and by chance if you ask the second question to the random-answere only, your chance of deciding the right path no more remains a certainty (as you cannot take a certain decision based on the random answer).
When determining necessary requirements of solving a problem by end state analysis, never think of HOW and concentrate fully on WHAT are the requirements at the End state,
In your riddle never think of how you would screen out the random-answerer by your first question, but know for sure that you have to eliminate him by your first question.
Okay, assume that you are able to achieve your objective of getting rid of random-answerer by your first question.
Solution to liar, truth-teller, random-answerer riddle - Stage 3: Determining form of a feasible second question
At this point you have ASSUMED that you have eliminated random-answerer and isolated the liar and truth-teller whom you would ask your second question.
Reality is virtual, but you have created a highly promising situation,
You have broken down the original puzzle into two parts and now solving the second part of a much simpler puzzle.
This is problem breakdown technique in action.
You know you have one helper who'll always answer truthfully to your question, and also you have the other who'll always reverse the truth and lie. No uncertainty of the random-answerer any more at this imagined stage.
As a start you ask a simple trial question to ANY OF THE TWO without much thinking,
Trial question 1: "Is the path on my left the safe path?".
As ANSWERS FROM BOTH OF THEM have to be analyzed for making the decision now (you don't know who is what), you would realize the unpleasant truth.
Conclusion 2: ANSWERS FROM THE TWO OF THEM to the simple question WILL SURELY BE OPPOSITE.
You won't be able to decide which is the safe path.
Try out any question similar to the question above and realize that asking a question of this form won't lead you to solution.
Realizing the limitation of this form of question, the critical question you face,
Critical Question 1: What is so special about this type of question?
Now you are not thinking about the CONTENT OF THE QUESTION, but you are thinking ABOUT THE FORM OF THE QUESTION, simply because all questions of this type have failed.
This is the point where you have to JUMP ACROSS A SHORT UNKNOWN GAP.
You may think desperately for days to understand what is so special about the form of the simple question and how the form may be changed for the right question and may not reach anywhere.
Or, you may realize in a flash that the word SIMPLE is the KEY. The key pattern of the question is,
Conclusion 3: It is a single and simple question.
And this form won't work.
What is to be done?
Conclusion 4: Just ask a COMPOUND QUESTION COMPRISING OF MORE THAN ONE QUESTION.
This is applying Problem solving technique of Changing the property of the main entity.
The main entity is the Question and its Property of number of component questions in the Question is changed from 1 to 2.
By the rules of English language you can join many single sentences together to form a compound sentence.
Having covered good ground with certainty now your attention shifts to the ANSWER itself.
Solution to liar, truth-teller, random-answerer riddle - Stage 4: Finalizing the requirement specification of the second question
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 again: 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,
Critical Question 2: What must be the NATURE OF THE ANSWER from the two for you to know the safe path?
You already know that the answer to a simple question from one helper will be NO and the other, YES, just the opposite.
So you are able to make a firm conclusion,
Conclusion 5: If the answers from the two are OPPOSITE, you won't find the safe path. Answer from both must be same.
There is no going away from this. The conclusion is actually a fact and an inviolable truth.
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 an important breakthrough.
Combining two results, you specify the requirements of the second question,
- The question must be a compound question comprising of more than one question, and,
- Answer to the question must be same if asked to BOTH OF THE HELPERS ISOLATED SIMULTANEOUSLY OR SEPARATELY.
Solution to the liar, truth-teller, random-answerer riddle - Stage 5: Finalizing the actual second question that will solve the riddle
With clear idea of the nature of the answer and form of the question, a possible safe question to ask would be,
Trial question 2: Are you the truth-teller and do you think the path to my left is the safe path?
This again is a conventional and the simple way to combine two questions.
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 SAME ANSWER BY BOTH.
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 this 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 on how to combine by analyzing the results.
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 would first combine the two to a final answer of "NO" and then by habit would reverse it to YES. HE would be 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 with certainty then,
One question must be DEPENDENT on the other so that in both possible scenarios for the liar, he would be forced to reverse the truth twice.
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 he has answered it in line with his habit would face the second DEPENDENT question and reverse answer to the first independent question once more by the second 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 liar must answer the FIRST of the two component questions following rules of the language,
'whether the "QUESTION" is true.'
The liar reverses the correct answer to this first component (and independent) 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 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 would finally form the single question that would lead you to know the safe path,
Second question solution: 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.
But your job is not finished.
First goal achieved, you take up the second challenge of SCREENING OUT THE RANDOM-ANSWERER BY YOUR FIRST QUESTION.
Do you realize that you are in fact going from End to Start, or back to front? This is the well-known powerful general problem solving technique of Working backwards approach.
Systematic solution to the liar, truth-teller, random answerer riddle - Stage 6: Screening out random-answerer by the first question
At the outset to this final stage you realize that you have to focus on the nature of the question that would produce the desired result.
As all three know about each other's nature of answering, and also as your objective in this situation is to screen out the random-answerer, the conclusion is easy to make,
Conclusion 6: The question must be on the nature of answering of the helpers.
It's an easy conclusion to make, but what should be the next decision about the first question to ask? That's not so easy to decide.
So you take up the third mental trial question. Pointing to one of the helpers, you ask another,
Trial question 3: Do you think he is the habitual liar?
Assuming that the questioned helper is the truth-teller and pointed helper is the liar, answer will be yes. But, if the pointed helper is the random-answerer answer will be no. And without knowing the nature of the questioned helper, you would be in deeper confusion.
But why is the confusion so much with the trial question of this form?
It is because the approach is too simple and,
Combination of possible outcomes are too many as you have covered only two of the three helpers asking the question on the nature of ONLY ONE OTHER HELPER not both.
A general principle of questioning
When you have a single question to ask, you must form the question that would reveal MAXIMUM INFORMATION ABOUT THE SUBJECTS.
If you think a bit, you would realize that this is the MOST EFFECTIVE APPROACH when you have only one arrow in your quiver of arrows. The arrow must be the most effective and most powerful arrow.
The critical question you raise to yourself,
Critical Question 3: What is the question that would reveal maximum information about the nature of answering of the helpers.
As you start thinking of possible answers to the leading question, you realize that you are faced again with a small gap that you have to jump across.
Common knowledge about revealing maximum information about subjects by a single question: COMPARE NATURE OF SUBJECTS
As you have to ask to any one of the three,
To cover maximum number of subjects, you have to include both the other two in your question, and,
To reveal maximum amount of information about the nature of answering of other two, you would ask the question ON COMPARING NATURE OF ANSWERING OF THE OTHER TWO.
As an example, without much thought you would form a trial question again that you ask to a FIRST HELPER,
Trial question 4: Is the THIRD HELPER (point out) more truthful than the SECOND HELPER (point out)?
As the trial question is indeed the most promising one fulfilling all requirements that we are aware of, we'll take the trouble to compile ALL possible answer scenarios in the following table.
When you scan through the possible answers in the six different combination scenarios, you realize that
Conclusion 8: For the final question, If you choose the SECOND HELPER for answer YES, you would surely eliminate Random-answerer for the first and third scenarios.
Conclusion 9: For the final question, if you choose the THIRD HELPER when answer is "NO" you would surely eliminate the random-answerer for the second and fourth scenarios.
But what about the last two scenarios?
Logic and Reason: As you are choosing the second helper for answer "YES" and the third helper for answer "NO", the first helper questioned random-answerer is automatically eliminated for the last two scenarios (in no case you are choosing the first helper).
Desired result is achieved by asking the first question to any First helper, "Is the THIRD HELPER (point out) more truthful than the SECOND HELPER (point out)?",
You would ask the second question to one of the helpers who may either be the liar or the truth-teller, as you wanted all along your reasoning.
And you have already decided the second question that you would ask now to any of the two helpers isolated,
Second question solution: What will be your answer if I ask you whether the path to my left is the safe path?
If the answer is YES, you would take the left path, and if it is NO, you take the right path.
And again you would be safe.
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.
- Problem breakdown technique: This a natural problem solving technique of splitting a large problem into smaller manageable problem chunks to solve pat by part and finally combine the results of the individual parts.
- 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 assessing promise of each change, often proves to be crucial in solving a complex problem. Great innovation can be created by this technique.
- 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.
- Working backwards approach: A powerful problem solving technique that proceeds from end to start.
- 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.
Puzzles you may enjoy
Logic analysis puzzles
Liar, Truth-teller, Random-answerer riddle