Problem 34 - Discrete Mathematics with Applications

Question

Four women, one of whom was known to have committed a serious crime, made the following statements when questioned by the police: (B. Bissinger, Parade Magazine, 1993 ) $$\begin{array}{ll}{\text { Fawn: }} & {\text { "Kitty did it" }} \\ {\text { Kitty: }} & {\text { "Robin did it." }} \\ {\text { Bunny: }} & {\text { "I didn't do it" }} \\ {\text { Robin: }} & {\text { "Kitty lied." }}\end{array}$$ If exactly one of these statements is true, identify the guilty woman.

Step-by-Step Solution

Answer

The guilty woman is Bunny, as her statement "I didn't do it" was proven to be false through a logical analysis of the given statements and problem constraints.

Step 1: Analyze Fawn's statement

Fawn says: "Kitty did it." We first assume Fawn's statement is true and the others are false. If Kitty is guilty and Fawn's statement is true, it implies that Robin's statement ("Kitty lied.") is also true. But this is a contradiction since only one statement can be true. Therefore, Fawn's statement is false.

Step 2: Analyze Kitty's statement

Kitty says: "Robin did it." Similar to the previous step, we assume Kitty's statement is true and the others are false. If Robin is guilty and Kitty's statement is true, it implies that Fawn's statement ("Kitty did it.") is false, which is valid. Therefore, Kitty's statement can be true.

Step 3: Analyze Bunny's statement

Bunny says: "I didn't do it." If Bunny's statement is true and the others are false, then Fawn's statement ("Kitty did it.") is false, and Kitty's statement ("Robin did it.") would also be false. Additionally, Robin's statement ("Kitty lied.") would be true as Kitty lied about Robin being guilty. That would mean that, in this case, two statements are true which contradicts the problem's requirement that only one statement is true. So, Bunny's statement is false, and Bunny is guilty.

Step 4: Determine the guilty woman

Based on the analysis in the previous steps, we found that Bunny is guilty. This means the only true statement is Robin's statement ("Kitty lied.") since it was never contradicted and is consistent with our analysis of the other statements. Therefore, the guilty woman is Bunny.

Key Concepts

Logical Reasoning

Logical reasoning is all about using given information to arrive at conclusions. In exercises like truth-tellers and liars puzzles, we use logic to determine who is telling the truth based on the condition that only one statement is true. We start by assuming each statement might be the true one and then see if this assumption leads to a contradiction or consistency with the rules given.

This approach requires a step-by-step analysis:

• Assume one statement is true.
• Check if this assumption makes all other statements false.
• Determine if these assumptions adhere to the given problem conditions.
• If a contradiction is found, that assumption is false.
• If consistent, the assumption might be true.

Using logical reasoning, we systematically examine each possibility, helping us find the truth effectively.

Truth-Tellers and Liars

Truth-tellers and liars puzzles are intriguing because they test our ability to distinguish truth from falsehood using minimal clues. In this scenario, we know that only one statement out of four is true, meaning the other three are lies.

Here is how we can dissect the responses:

• Understand that a truth-teller gives a statement consistent with reality.
• A liar's statement contradicts the reality.
• Based on the truth-teller's one true statement, identify inconsistencies among other statements.

This type of puzzle challenges you to think critically about the relationships between statements and their meanings.

Problem-Solving

Problem-solving in logic puzzles often involves a process of elimination backed by assumptions. With the truth-tellers and liars puzzle, you systematically test each statement to see which fits the criteria of having exactly one true statement.

Steps for solving:

• Consider each statement potentially true and assess the implications.
• Exclude those that result in more than one statement being true.
• Find the scenario where the truth-speaker’s statement aligns with the puzzle's constraints.

By following these steps meticulously, you narrow down the possibilities to find the correct answer. This reinforces skills such as critical analysis and logical deduction.

Contradictions in Statements

In logic puzzles, contradictions help us identify false statements. A contradiction occurs when an assumption leads to inconsistencies. In the problem of determining who committed the crime, each statement needs to be evaluated:

• Assuming one woman's guilt checks if any statement contradicts this implication.
• If an assumption creates more truths than allowed, return to examine contradictions.
• Statements that don't produce contradictions in the context of one truth indicate inconsistencies.

By exploring contradictions, we uncover logical inconsistencies, which help us rule out incorrect assumptions and lead to the correct conclusion: Bunny being guilty.