# Episode 3: Refining the Toolkit

**Objective:** To master advanced maneuvers—including case analysis, symmetry, and uniqueness—and to understand the rigorous logic of disproof.

### I. Introduction: Expanding the Arsenal

* **Beyond the "Big Three":** Moving past the straightforward Direct, Contrapositive, and Contradiction methods.

* **Nuanced Logic:** Why some theorems require splitting the problem into pieces or proving that only one possible solution exists.

### II. Segment 1: Divide and Conquer (Proof by Cases)

* **The Logical Structure:**

* **The Requirement of Exhaustiveness:** Ensuring the chosen cases cover every possible scenario in the universe of discourse.

* **Common "Splits":**

* **Integers:** Parity (even vs. odd) or remainders modulo N.

* ** ...