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Chapter 1: Problem 20

An explorer is captured by a group of cannibals. There are two types of cannibals鈥攖hose who always tell the truth and those who always lie. The cannibals will barbecue the explorer unless he can determine whether a particular cannibal always lies or always tells the truth. He is allowed to ask the cannibal exactly one question. a) Explain why the question 鈥淎re you a liar?鈥� does not work. b) Find a question that the explorer can use to determine whether the cannibal always lies or always tells the truth.

Short Answer

Expert verified
The question 'Are you a liar?' does not work because both would answer 'no'. The explorer should ask, 'If I asked if 2 + 2 = 4, would you say yes?'.

Step by step solution

01

- Understanding the Problem

The explorer is allowed to ask only one question to a cannibal and determine if the cannibal always lies or always tells the truth. The cannibal either always lies or always tells the truth.
02

- Analyze Question: 'Are you a liar?'

If the explorer asks, 'Are you a liar?', both types of cannibals will say 'No'. The truthful cannibal says 'No' because they are truthful and not liars. The lying cannibal says 'No' because they always lie, so they will not admit to being a liar either. Thus, this question does not help determine the cannibal's nature.
03

- Finding a New Question

To find a question that works, consider a question whose truthfulness depends on an objective fact, regardless of who answers it. The question the explorer can ask is: 'If I asked you whether you always tell the truth, would you say yes?'
04

- Evaluating the New Question

Now evaluate the responses:- If the cannibal is truthful, they would honestly answer 'yes' to 'Do you always tell the truth?'. Thus, when asked, 'If I asked you whether you always tell the truth, would you say yes?', the truthful cannibal will say 'yes'.- If the cannibal is a liar, they would lie and answer 'no' to 'Do you always tell the truth?'. Since they are a liar, to the new question 'If I asked you whether you always tell the truth, would you say yes?', they will also lie and say 'yes'.
05

- Conclusion

Both honest and dishonest cannibals saying 'yes' to the new question signifies their nature. However, for the explorer to clearly determine the nature, use the question: 'If I asked you if 2 + 2 = 4, would you say yes?'- The truthful cannibal will answer 'yes' because 2 + 2 does equal 4 and they tell the truth.- The lying cannibal will answer 'no' because if asked whether 2 + 2 = 4, they would lie and say 'no'.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

logic puzzles
Logic puzzles challenge our reasoning and problem-solving skills. They often present scenarios where we must use our understanding of logic to find a solution. In the 'cannibal problem,' the explorer faces a classic logic puzzle. He must determine whether a cannibal always tells the truth or always lies by asking only one question.
The key to solving logic puzzles like this one is to analyze the rules and constraints given. Here, the puzzle hinges on finding a question whose answer reveals the cannibal's true nature. By understanding how each type of cannibal would respond, the explorer can deduce the correct answer.
These types of puzzles are not only fun but also sharpen critical thinking and logical deduction skills.
truth-tellers and liars
Truth-tellers always speak the truth, while liars always lie. This binary behavior is crucial in solving the 'cannibal problem.' Both types of cannibals will react differently to questions based on their nature. For example, if asked 'Are you a liar?', both types would say 'No.'
Truth-tellers' responses can be trusted because they reflect reality. Conversely, liars' responses are intentionally false. Understanding this dichotomy helps in framing questions that can reveal the truth. Consider the question 'If I asked you whether you always tell the truth, would you say yes?' This question exploits the predictable behavior of truth-tellers and liars.
- Truth-teller would truthfully say 'yes.'
- Liar would falsely say 'yes,' making their lie evident.
By focusing on questions that expose the underlying truth through their lies, we can easily identify the liar.
decision problems
Decision problems require making a choice based on the information available. In the cannibal puzzle, the explorer faces such a decision problem. He must decide what question to ask to distinguish between the two types of cannibals.
To solve decision problems, it鈥檚 important to:
- Understand the constraints and rules.
- Evaluate potential outcomes of different choices.
In this problem, evaluating the question 'Are you a liar?' leads to the realization that it doesn鈥檛 solve the problem because both cannibals respond 'No.'
Through careful analysis, the explorer finds the effective question: 'If I asked you if 2 + 2 = 4, would you say yes?' The responses to this question help make the correct decision:
- Truth-teller says 'yes' because it鈥檚 true.
- Liar says 'no' because they lie about the truth.
logical deduction
Logical deduction is the process of reasoning from one or more statements (premises) to reach a logically certain conclusion. In solving the cannibal problem, the explorer uses logical deduction to determine the nature of the cannibal.
Here's how logical deduction works in this scenario:
- Premise: There are truth-tellers and liars.
- Premise: Truth-tellers answer truthfully; liars do not.
By asking a question based on a known fact (e.g., 'If I asked you if 2 + 2 = 4, would you say yes?'), the explorer can deduce the following:
- Truth-teller would affirm the truth and say 'yes.'
- Liar, aiming to lie, would actually confirm their deceit by saying 'no.'
This process demonstrates how logical deduction guides us through steps to reach a reliable conclusion, solving the puzzle effectively and efficiently.

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Most popular questions from this chapter

Exercises \(40-44\) deal with the translation between system specification and logical expressions involving quantifiers. Translate these specifications into English, where \(F(p)\) is "printer \(p\) is out of service," \(B(p)\) is "Printer \(p\) is busy," \(L(j)\) is "Print job \(j\) is lost, " and \(Q(j)\) is "Print job \(j\) is queued." $$ \begin{array}{l}{\text { a) } \exists p(F(p) \wedge B(p)) \rightarrow \exists j L(j)} \\ {\text { b) } \forall p B(p) \rightarrow \exists j Q(j)} \\\ {\text { c) } \exists j(Q(j) \wedge L(j)) \rightarrow \exists p F(p)} \\\ {\text { d) }(\forall p B(p) \wedge \forall j Q(j)) \rightarrow \exists j L(j)}\end{array} $$

Determine the truth value of each of these statements if the domain for all variables consists of all integers. $$ \begin{array}{ll}{\text { a) } \forall n\left(n^{2} \geq 0\right)} & {\text { b) } \exists n\left(n^{2}=2\right)} \\ {\text { c) } \forall n\left(n^{2} \geq n\right)} & {\text { d) } \exists n\left(n^{2}<0\right)}\end{array} $$

Find a counterexample, if possible, to these universally quantified statements, where the domain for all variables consists of all integers. a) \(\forall x \forall y\left(x^{2}=y^{2} \rightarrow x=y\right)\) b) \(\forall x \exists y\left(y^{2}=x\right)\) c) \(\forall x \forall y(x y \geq x)\)

Prove or disprove that you can use dominoes to tile the standard checkerboard with two adjacent corners removed (that is, corners that are not opposite).

Suppose that Prolog facts are used to define the predicates mother \((M, Y)\) and father \((F, X),\) which represent that \(M\) is the mother of \(Y\) and \(F\) is the father of \(X\) , respectively. Give a Prolog rule to define the predicate \(\operatorname{sibling}(X, Y)\) which represents that \(X\) and \(Y\) are siblings (that is, have the same mother and the same father).
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