Liar Puzzle Reasoning

Let $A$, $B$, and $C$ represent the statements "A is a truth-teller", "B is a truth-teller", and "C is a truth-teller", respectively. Correspondingly, $¬A$, $¬B$, and $¬C$ represent "A is a liar", "B is a liar", and "C is a liar". We assume each person is either always a truth-teller or always a liar.

The given statements are:

• A claims: "$B$ is a liar and $C$ is a liar" (i.e., $¬B \land ¬C$)
• B claims: "$C$ is a liar" (i.e., $¬C$)

Analyzing A's Status

We test the possibilities for A:

• Case 1: Assume A is a truth-teller ($A$).
  • If $A$ is true, A's statement ($¬B \land ¬C$) must be true. This means $B$ is a liar ($¬B$) and $C$ is a liar ($¬C$).
  • Since $B$ is a liar ($¬B$), B's statement ($¬C$) must be false.
  • If B's statement ($¬C$) is false, it implies $C$ is true (C is a truth-teller).
  • This leads to a contradiction: $C$ must be a liar (from A's statement) and $C$ must be a truth-teller (from B's lie).
  • Therefore, the assumption that A is a truth-teller is false.
• Conclusion on A: A must be a liar ($¬A$).

Analyzing B's and C's Status

Since A is a liar ($¬A$), A's statement ($¬B \land ¬C$) is false. The negation, $¬(¬B \land ¬C)$, must be true. By De Morgan's laws, this is equivalent to $B \lor C$, meaning at least one of B or C is a truth-teller.

Now, let's consider B's statement ($¬C$) based on whether B is a truth-teller or a liar:

• Possibility 1: B is a truth-teller ($B$).
  • B's statement ($¬C$) is true. So, C is a liar ($¬C$).
  • Current status: $¬A$, $B$, $¬C$. (A is liar, B is truth-teller, C is liar).
  • This scenario is consistent: A is a liar, B tells the truth (about C being a liar), and C is a liar. There are exactly two liars (A and C).
• Possibility 2: B is a liar ($¬B$).
  • B's statement ($¬C$) is false. So, C is a truth-teller ($C$).
  • Current status: $¬A$, $¬B$, $C$. (A is liar, B is liar, C is truth-teller).
  • This scenario is consistent: A is a liar, B lies (about C being a liar, which is true), and C is a truth-teller. There are exactly two liars (A and B).

Determining the False Statement

We have two possible valid scenarios:

1. A is Liar, B is Truth-teller, C is Liar (Total 2 liars)
2. A is Liar, B is Liar, C is Truth-teller (Total 2 liars)

Let's evaluate the given options based on these scenarios:

• Option 1: A is a liar ($¬A$)
  • This is true in both scenarios.
• Option 2: Exactly one of B and C is a liar
  • Scenario 1: C is a liar, B is not. (True)
  • Scenario 2: B is a liar, C is not. (True)
  • This statement is true in both scenarios.
• Option 3: Exactly one among A, B and C is a liar
  • Scenario 1: A and C are liars (2 liars). (False)
  • Scenario 2: A and B are liars (2 liars). (False)
  • This statement is false in both scenarios.
• Option 4: Exactly two among A, B and C are liars
  • Scenario 1: A and C are liars. (True)
  • Scenario 2: A and B are liars. (True)
  • This statement is true in both scenarios.

The question asks for the statement that is FALSE. Based on the analysis, the statement "Exactly one among A, B and C is a liar" is false.