Water. Earth. Fire. Air. Long ago, the four nations lived together in harmony. Then,
everything changed when the Fire Nation attacked. Only the Alex-tars, masters of all four
elements, could stop them, but when the world needed them most, they vanished. A hundred
years pass and you discover a group of 160 benders who can each bend at least one of the four
elements: 124 can bend water, 113 can bend earth, 127 can bend fire, and 119 can bend air.
Prove that there exist at least three Alex-tars in the group of 160 benders to restore peace and
unity to the world!

This problem can be approached using the **principle of inclusion-exclusion** and **Pigeonhole Principle** to prove that there must be at least three "Alex-tars"—benders who can bend all four elements.

### Step 1: Understand the Setup
We are given the following:

- Total benders: 160
- Benders who can bend water: 124
- Benders who can bend earth: 113
- Benders who can bend fire: 127
- Benders who can bend air: 119

The goal is to prove that there exist at least three individuals who can bend **all four** elements.

### Step 2: Overlap of Sets
Let’s define the following sets:
- \( W \):

Water. Earth. Fire. Air. Long ago, the four nations lived together in harmony. Then, everything changed when the Fire Nation attacked. Only the Alex-tars, masters of all four elements, could stop them, but when the world needed them most, they vanished. A hundred years pass and you discover a group of 160 benders who can each bend at least one of the four elements: 124 can bend water, 113 can bend earth, 127 can bend fire, and 119 can bend air. Prove that there exist at least three Alex-tars in the group of 160 benders to restore peace and unity to the world!

Learn how to prove that there are at least three Alex-tars—benders who can control all four elements—in a group of 160 benders. This problem involves advanced mathematical concepts such as the principle of inclusion-exclusion, set theory, and the pigeonhole principle. Explore how to analyze overlapping sets of benders who can control water, earth, fire, and air. Suitable for advanced high school students (Grades 11-12).

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