What are the theorems in mathematics which can be proved using completely different ideas? I would like to know about theorems  which can give different proofs using completely different techniques.  

Motivation:
  When I read from the book Proof from the Book, I saw there were many many proof for the same theorem using completely different fields of mathematics.
  (There were Six proofs of the infinity of primes based on different ideas even using topology.)

I wonder How mathematicians give this kind of proofs.
If you know such theorems and proofs please share them with me. Thank you. 

A large rectangle is tiled with smaller rectangles. Each of the smaller rectangles has at least one integer side. Must the large rectangle have at least one integer side?
What if you replace 'integer' with 'rational' or 'algebraic'?
Here are fourteen proofs.
