Question

Solve the following "Liars" puzzle (from Phillips 1934): Five schoolgirls sat for an examination. Their parents-so they thoughtshowed an undue degree of interest in the result. They therefore agreed that, in writing home about the examination, each girl should make one true statement and one untrue one. The following are the relevant passages from their letters: \- Betty: "Kitty was second in the examination. I was only third." \- Ethel: "You'll be glad to hear that I was on top. Joan was second." \- Joan: "I was third, and poor old Ethel was bottom." \- Kitty: "I came out second. Mary was only fourth." \- Mary: "I was fourth. Top place was taken by Betty." What in fact was the order in which the five girls were placed?

Short answer

The order is: Mary first, Betty second, Joan third, Kitty fourth, and Ethel fifth.

Step-by-step solution

Analyze Betty's Statements

Betty claims that Kitty was second, and she herself was third. If Betty is telling the truth about Kitty being second, then her other statement that she was third must be false. Therefore, Betty cannot be third.

Analyze Ethel's Statements

Ethel claims she is on top and Joan was second. If Ethel is telling the truth about being on top, the Joan second statement should be false. Thus, Joan cannot be second if Ethel is telling the truth about being top.

Analyze Joan's Statements

Joan claims she was third and Ethel was bottom. If Joan is telling the truth about being third, then Ethel cannot be bottom. Alternatively, if Ethel is bottom, Joan cannot be third.

Analyze Kitty's Statements

Kitty claims she was second and Mary was fourth. If Kitty is really second, then her statement about Mary being fourth is false, meaning Mary can be any position except fourth. Thus, Mary is not fourth.

Analyze Mary's Statements

Mary claims she was fourth, and Betty was on top. If Mary is genuinely fourth, then Betty being top is false. This disproves Ethel's claims about being top.

Deduce the True Positions

To maintain consistency among all the statements and the logic deduced: - If Joan is telling the truth about Ethel being bottom, then Ethel is indeed last. - If this is true, then based on the steps, the order falls out as: Ethel fifth, Mary fourth, Joans third. - Since Kitty, based on Betty’s claim (one of these two is partially correct due to Mary’s claim disproving Kitty’s claim about Mary's rank), can’t be second due to Mary’s contradictions, thus Betty is second and Kitty ends up fifth.

Final Order Verification

Based on the deduction: - Top position: Mary (She lied about being fourth and Betty being top) - Betty lied only about her own position being third when it was genuinely second - Based on Joan’s bottom claim about Ethel (truthfully third with a lie about position) - Confirm positions: Mary (1st), Betty (2nd), Joan (3rd), Kitty (4th), Ethel(5th) - Bet for Kitty and Ethel’s clues lies in the configuration on third and order between 1-4 on execution.

Key concepts

Deductive Reasoning

Deductive reasoning is a logical process where conclusions are derived from a general set of premises or a general rule. In the context of the 'Liars' puzzle, deductive reasoning helps piece together the truth from each girl's conflicting statements.
Each schoolgirl provides one truthful statement and one false one, which is a classic setup for using deduction.

Here are some steps of how deductive reasoning is applied in logic puzzles like this:

• Identify all given statements and their possible veracity (truth or falsehood).
• Analyze what each statement would imply if taken as true.
• Consider contradictions that arise when certain statements are assumed to be true.
• Use the process of elimination, where identifying a false statement gives clues about the others.

This method allows us to eliminate inconsistent positions for each girls' rankings, thereby revealing the true order. This is how the puzzle solver deduced that the order was Mary first, Betty second, Joan third, Kitty fourth, and Ethel fifth.

Problem Solving

Problem solving involves tackling complex issues by breaking them down into manageable parts, making logic puzzle-solving a great exercise for this skill. In the 'Liars' puzzle, this skill is crucial as we dissect each statement to reveal the accuracy of the girls' rankings.

Approaching such a problem effectively involves several strategies:

• Working Methodically: Analyze each statement separately to maintain clarity and organization.
• Using Process of Elimination: Once a statement is determined to be false, the other linked statement is information that can be trusted.
• Checking Consistency: Ensure that all facts fit logically without contradictions.
• Rechecking Conclusions: Verify each step to ensure that the logical deductions align with the problem's requirements.

These techniques allow solvers to systematically determine that Mary eventually becomes the top position holder despite declaring herself fourth, highlighting the importance of structured problem solving.

Critical Thinking

Critical thinking is about being objective and making decisions based on well-evaluated reasoning. It involves analysis, reflection, and the ability to evaluate an argument or a set of claims, especially in the complex web created in logic puzzles like the 'Liars' puzzle.

When engaging in critical thinking for problem solving:

• Question Assumptions: Consider that not all presented information is true, such as in the case of the statements by each girl in the puzzle.
• Prioritize Information: Weigh the possibilities of each statement before and after new truths are discovered to navigate the problem efficiently.
• Adjust Reasoning: As more information is processed, adjust your understanding and approach according to logical deductions and newfound truths.
• Reflect on Conclusions: After reaching a solution, reflect on the steps taken to understand the process and its correctness.

In this particular puzzle, critical thinking was needed to understand why certain misleading statements couldn't be true, leading to the resolution and correct rank order of Ethel being fifth and Mary as the unexpected summit of the list.