Logical Puzzlers

In solving logic puzzles involving truth-tellers and liars, students should employ the principle of the law of the excluded middle, which asserts that every statement must be either true or false, with no third option. This principle is fundamental in binary logic and forces us to categorize each person in the puzzle as either a truth-teller, who always speaks truthfully, or a liar, who always lies.

To solve the puzzle, students must methodically test assumptions about whether each individual is a truth-teller or liar, systematically verifying whether these assumptions lead to a consistent interpretation of all statements. By working through the puzzle step by step, rejecting any assumptions that create contradictions, students will ultimately find the only combination of truth-tellers and liars that satisfies the entire set of claims.

Puzzle:
You are on an island where there are only two types of inhabitants: truth-tellers and liars. You meet two inhabitants, A and B. A says, “B is a liar.” B says, “A is a truth-teller.”

Question:
What are A and B?

Solution:

1. Assume A is a truth-teller.

   • If A is a truth-teller, then A’s statement must be true. A said, “B is a liar.”
   • Therefore, B must be a liar.

2. Now check if B being a liar is consistent with B’s statement.

   • B, as a liar, would always lie. B said, “A is a truth-teller.”
   • Since B is a liar, B’s statement must be false. Therefore, A is not a truth-teller. But this contradicts our assumption that A is a truth-teller.

3. So, A cannot be a truth-teller. Therefore, A must be a liar.

4. If A is a liar, then A’s statement is false.

   • A said, “B is a liar.” Since A is a liar, this statement is false, meaning B is actually a truth-teller.

5. Conclusion: A is a liar, and B is a truth-teller.

Puzzle:
You meet three inhabitants, C, D, and E.

• C says, “D is a liar.”
• D says, “E is a liar.”
• E says, “C and D are both liars.”

Question:
Who are the truth-tellers, and who are the liars?

Solution:

1. Assume C is a truth-teller.

   • If C is a truth-teller, then C’s statement, “D is a liar,” is true.
   • Therefore, D is a liar.

2. Now check D’s statement.

   • D, being a liar, must be lying when they say, “E is a liar.”
   • Since D is lying, it means that E is actually a truth-teller.

3. Now check E’s statement.

   • E, as a truth-teller, must be telling the truth when they say, “C and D are both liars.”
   • But this contradicts our assumption that C is a truth-teller.

4. Therefore, C cannot be a truth-teller. C must be a liar.

5. If C is a liar, then C’s statement, “D is a liar,” is false.

   • This means D is actually a truth-teller.

6. Now check D’s statement.

   • D, being a truth-teller, must be telling the truth when they say, “E is a liar.”
   • Therefore, E is a liar.

7. Conclusion: C and E are liars, and D is a truth-teller.

Puzzle:
You meet two inhabitants, F and G.

• F says, “Either I am a liar or G is a truth-teller.”

Question:
What are F and G?

Solution:

1. Break down F’s statement.

   • F says, “Either I am a liar or G is a truth-teller.”
   • This is a disjunction (either-or). For a disjunction to be false, both parts of it must be false. For it to be true, at least one part must be true.

2. Assume F is a truth-teller.

   • If F is a truth-teller, then F’s statement must be true. This means that either F is a liar or G is a truth-teller.
   • But F can’t be a liar if F is a truth-teller. So, the only possibility left is that G must be a truth-teller.

3. Check if this assumption holds.

   • If G is a truth-teller, then F’s statement is consistent with the assumption, since one part of the disjunction (G being a truth-teller) is true.

4. Conclusion: F is a truth-teller, and G is also a truth-teller.

Puzzle:
You come across two doors: one leads to safety, and the other leads to certain danger. There are two guards, H and I, one of whom always tells the truth, and the other always lies. You are allowed to ask only one yes-or-no question to one of the guards.

Question:
What single question should you ask to determine which door leads to safety?

Solution:

1. Key insight:

   • If you ask a truth-teller what the liar would say, the truth-teller will truthfully report the liar’s false statement.
   • If you ask a liar what the truth-teller would say, the liar will lie about the truth-teller’s truthful statement.
   • In either case, you will get the wrong answer about which door leads to safety.

2. The question to ask:

   • Ask either guard, “If I asked the other guard which door leads to danger, what would they say?”
   • No matter whom you ask, the answer you get will indicate the door that leads to safety.

3. Why this works:

   • If you ask the truth-teller, they will tell you the liar’s false answer.
   • If you ask the liar, they will lie about the truth-teller’s correct answer.
   • In either case, the guard will point to the dangerous door, so you should choose the opposite door.

4. Conclusion:

   • Ask either guard the question above, and then choose the opposite door to the one they point to.

Puzzle:
You meet four people, J, K, L, and M.

• J says, “K is a liar.”
• K says, “M is a truth-teller.”
• L says, “K is telling the truth.”
• M says, “J is a liar.”

Question:
Who is the truth-teller, and who are the liars?

Solution:

1. Assume J is a truth-teller.

   • If J is a truth-teller, then J’s statement, “K is a liar,” is true.
   • So, K must be a liar.

2. Now check K’s statement.

   • K, being a liar, would be lying about M being a truth-teller.
   • Therefore, M must be a liar.

3. Now check L’s statement.

   • L says, “K is telling the truth.” Since K is a liar, L’s statement must be false.
   • Therefore, L is also a liar.

4. Check M’s statement.

   • M, being a liar, says, “J is a liar.” This must be false, so J is indeed a truth-teller, confirming our initial assumption.

5. Conclusion: J is the only truth-teller, and K, L, and M are liars.

Puzzle:
Five people, N, O, P, Q, and R, are having a discussion. One of them is a truth-teller, and the rest are liars.

• N says, “O is a liar.”
• O says, “R is a truth-teller.”
• P says, “I am the truth-teller.”
• Q says, “P is lying.”
• R says, “N is telling the truth.”

Question:
Who is the truth-teller?

Solution:

1. Assume N is the truth-teller.

   • If N is the truth-teller, then N’s statement, “O is a liar,” is true.
   • Therefore, O is a liar.

2. Now check O’s statement.

   • O, being a liar, would be lying when they say, “R is a truth-teller.”
   • Therefore, R must be a liar.

3. Now check R’s statement.

   • R, being a liar, would be lying when they say, “N is telling the truth.”
   • But this contradicts our assumption that N is the truth-teller.

4. Therefore, N cannot be the truth-teller.

5. Assume R is the truth-teller.

   • If R is the truth-teller, then R’s statement, “N is telling the truth,” must be true.
   • Therefore, N is also telling the truth, so N is not a liar.

6. Now go back to N’s statement.

   • N says, “O is a liar.” This must be true because N is telling the truth.
   • Therefore, O is a liar.

7. Now check O’s statement.

   • Since O is a liar, O’s statement, “R is a truth-teller,” would be false, but that doesn’t cause any contradiction because R is indeed the truth-teller.

8. **Conclusion: R is the truth-teller, and everyone else (N, O

, P, and Q) are liars.**

These step-by-step solutions show how to rigorously apply logic without needing clever tricks. Each assumption is carefully checked against all the statements to eliminate contradictions and reach a sound conclusion.