 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 $F_n$ – the n-th Fibonacci Number.
We will also discuss some identities related to the Fibonacci Numbers using this idea of the Tiling.

For eg: $F_0 + F_1 + ... + F_n = F_{n+2} - 1$.
• 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

1. all the points lie on a single line; or
2. 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.