Every week we dedicate an hour to Beautiful Mathematics – the Mathematics that shows us how Beautiful is our Intellect. Today we are going to discuss the Fermat’s Little Theorem.

This week, I decided to do three beautiful proofs in this one-hour session…

**Proof of Fermat’s Little Theorem**( via Combinatorics )

It uses elementary counting principles. Here is a Wikipedia resource to learn it.

**Fibonacci Tilling**

This is about how we will prove that the number of ways we can cover a 2xn board with a 2×1 dominoes is the same as – the n-th Fibonacci Number.

We will also discuss some identities related to the Fibonacci Numbers using this idea of the Tiling.

For eg: .

**Sylvester – Gallai Theorem**and**Extremal Principle**

Here, we will show the proof of the theorem which states that

The **Sylvester–Gallai theorem** in geometry states that, given a finite number of points in the Euclidean plane, either

- all the points lie on a single line; or
- there is at least one line which contains exactly two of the points

This will introduce the students to the idea of the Extremal Principle and the Well Ordering Principle.

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