Stage: 2, 3, 4 and 5 Challenge Level: 
As suggested in the question it helps to start with simple
cases.
Algorithms. Stage 5 needs dealing with. Fibonacci sequence. Patterned numbers. Mathematical reasoning & proof. Mathematical induction. Codes and cryptography. Networks/Graph Theory. Combinatorics. Dynamical systems.
Published June 2000,January 2007.
Copyright © 2007. University of Cambridge. All rights reserved.
NRICH is part of the family of activities in the Millennium Mathematics Project, which also includes the Plus and Motivate sites.
email: nrich@damtp.cam.ac.uk