Problem 35 - 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 false, identify the guilty woman.

Step-by-Step Solution

Answer

The guilty woman is Kitty.

Step 1: Analyze Fawn's statement

: Fawn says: "Kitty did it." If this statement is true, Kitty is the guilty woman. Let's see if it's compatible with the other statements. - Kitty says, "Robin did it." This statement should be false. - Bunny says, "I didn't do it." This statement should be true. - Robin says, "Kitty lied." This statement should be true. In this scenario, there are two false statements (Kitty's and Robin's statements). Since there can only be one false statement, Fawn's statement must be false.

Step 2: Analyze Kitty's statement

: Kitty says: "Robin did it." If this statement is true, Robin is the guilty woman. Let's see if it's compatible with the other statements. - Fawn says, "Kitty did it." This statement should be false. - Bunny says, "I didn't do it." This statement should be true. - Robin says, "Kitty lied." This statement should be false. In this scenario, there are two false statements (Fawn's and Robin's statements). Since there can only be one false statement, Kitty's statement must be false.

Step 3: Analyze Bunny's statement

: Bunny says: "I didn't do it." If this statement is true, either Kitty, Fawn, or Robin would be the guilty woman. Let's see if it's compatible with the other statements. - Fawn says, "Kitty did it." This statement should be true. - Kitty says, "Robin did it." This statement should be false. - Robin says, "Kitty lied." This statement should be true. In this scenario, there is only one false statement (Kitty's statement). So, it's possible that Bunny's statement is true.

Step 4: Analyze Robin's statement

: Robin says: "Kitty lied." If this statement is true, it means that Kitty didn't tell the truth when she said, "Robin did it." Therefore, Robin is not the guilty woman. Let's see if it's compatible with the other statements. - Fawn says, "Kitty did it." This statement should be true. - Kitty says, "Robin did it." This statement should be false. - Bunny says, "I didn't do it." This statement should be true. In this scenario, there is only one false statement (Kitty's statement). So, it's also possible that Robin's statement might be true.

Step 5: Conclusion

: We have analyzed each woman's statement and identified which combinations result in having just one false statement. It turns out that the only scenario with exactly one false statement is while considering Bunny's statement to be true. Therefore, we have: - Fawn says, "Kitty did it." This statement is true. - Kitty says, "Robin did it." This statement is false. - Bunny says, "I didn't do it." This statement is true. - Robin says, "Kitty lied." This statement is true. The guilty woman is Kitty.

Key Concepts

Logical Reasoning

Logical reasoning is a process that involves using a structured approach to solve problems or make decisions. Here, we are using logical reasoning to identify who among four women is guilty. Each woman made a statement, and we need to determine which of these is false given the constraint that exactly one statement is false. This type of reasoning often involves:

• Evaluating each statement one by one and understanding the implications of each being either true or false.
• Analyzing how the truth or falsity of one statement affects the others.
• Using deductive reasoning to eliminate impossibilities until arriving at a consistent solution.

Logical reasoning is often used to derive conclusions based on given premises, and recognizing the connections between these premises helps us arrive at the right answer.

Truth Tables

Truth tables are a helpful tool for organizing the different scenarios that might arise from different truth values of statements. In our exercise, each woman makes a statement about who committed the crime. By evaluating these statements, we can systematically determine which statement is false. For each statement:

• Assess whether the statement could be true or false.
• If we assume one statement is true, determine which others must then be false to keep the count to one false statement.
• Create a table-like structure in your mind (or on paper) to keep scenarios organized.

This kind of structured evaluation helps visualize all logical possibilities at once. By deducing which combination contains exactly one false statement, we can accurately solve logical puzzles like the one presented.

Logical Statements

A logical statement is a sentence that can be clearly classified as true or false. In logical puzzles, these often take the form of declarations like the ones made by the women in the original exercise. Let's consider the nature of logical statements:

• Statements have clear truth values; meaning each is either true or false.
• They often assert something about the relationships between objects or events.
• Within puzzles, they work together to create a system of interdependent truths and lies.

In practice, logical statements require us to critically analyze their meanings and implications. By unearthing contradictions and consistencies, we find solutions and make sense of the logical framework. Simply, they direct us as we resolve problems, like determining who among the women made the false statement and, consequently, who is guilty.