**Correct Answer: D. Alternator, Liar, Truth-teller** ### **Step-by-Step Solution** To solve high-complexity Binary Logic puzzles, we use **Contradiction and Consistency Mapping**. We test the identity of the speakers by making an assumption and checking for logical consistency. **Step 1: Analyzing $B$’s 3rd Statement and $A$’s 2nd Statement** Note that $B3$ says "$A2$ is a lie." * If $B$ is the Truth-teller, then $A2$ must be a **Lie**. * If $B$ is the Liar, then $B3$ is false, meaning $A2$ must be **True**. **Step 2: Testing Assumption 1 — $A$ is the Truth-teller** If $A$ is the Truth-teller: 1. $A1$ ("$C$ is the Liar") must be **True**. 2. $A2$ ("I am the Alternator") must be **True**. **Contradiction:** $A$ cannot be the Truth-teller and the Alternator simultaneously. *Therefore, $A$ is NOT the Truth-teller.* **Step 3: Testing Assumption 2 — $C$ is the Truth-teller** If $C$ is the Truth-teller, all his statements must be **True**: 1. $C3$: "$A$ is the Alternator" $\rightarrow$ **True**. 2. Since $A$ is the Alternator and $C$ is the Truth-teller, $B$ must be the **Liar**. 3. $C2$: "$B$ is the Liar" $\rightarrow$ **True** (Consistent). 4. $C1$: "If $A$ is the Truth-teller, then $B$ is the Alternator." * In logic, a conditional $P \rightarrow Q$ is **True** if $P$ is **False**. * Since $A$ is not the Truth-teller, the statement is vacuously **True** (Consistent). Now, let's verify if this fits with $A$ as the Alternator and $B$ as the Liar: **Verifying $B$ as the Liar (All must be False):** 1. $B1$: "$A$ is the Liar" $\rightarrow$ **False** (Since $A$ is the Alternator). 2. $B2$: "$C$ made exactly two false statements" $\rightarrow$ **False** (Since $C$ is the Truth-teller, he made zero false statements). 3. $B3$: "$A2$ is a lie" $\rightarrow$ **False** (This implies $A2$ must be **True**). **Verifying $A$ as the Alternator (Statements must alternate $T-F-T$ or $F-T-F$):** 1. $A1$: "$C$ is the Liar" $\rightarrow$ **False** (Since $C$ is the Truth-teller). 2. $A2$: "$I$ am the Alternator" $\rightarrow$ **True** (Since our assumption holds). 3. $A3$: "$B$ always tells the truth" $\rightarrow$ **False** (Since $B$ is the Liar). *Pattern for $A$: $(F - T - F)$. This is a valid Alternator sequence.* **Step 4: Final Identity Map** The identities are consistent with all statements: $$\begin{array}{|c|c|c|c|c|} \hline \textbf{Person} & \textbf{Statement 1} & \textbf{Statement 2} & \textbf{Statement 3} & \textbf{Identity} \\ \hline A & \text{False} & \text{True} & \text{False} & \text{Alternator} \\ \hline B & \text{False} & \text{False} & \text{False} & \text{Liar} \\ \hline C & \text{True} & \text{True} & \text{True} & \text{Truth-teller} \\ \hline \end{array}$$ Thus, the sequence $(A, B, C)$ is **(Alternator, Liar, Truth-teller)**.