An Hour of Beautiful Proofs

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 2x1 dominoes is the same as $latex 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: $latex F_0 + F_1 + ... + F_n = F_{n+2} - 1$.

• 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.