At play

I didn't begin to enjoy, let alone love, mathematics until I started to play around with a few problems that caught my interest. It wasn't a serious research effort at first - it was for just for fun. I wanted to explore things for myself.

I am addicted to what Richard Feynman called "the pleasure of finding things out." I have had a great time exploring things and I hope to have a great time sharing what I've found. My work has opened some windows for me into worlds of astonishing beauty; I can only hope that some others will enjoy similar encounters.

For several years, I have wanted to write a series of books named At play that present some of my research in math and algorithms, especially relating to graph theory. I want to invite people to have fun with math and to learn and do original research through play. It happens that most of the problems that I've worked on, my approaches to them, and many of the results that I've found are easily understandable by people who don't have a math background.

I've been trying to find my ideal reader for these books. I recently visited one of my former students and his wife and was introduced to their two children, Cameron and Gillian, near and early teens respectively. They are both gifted and great kids. Cameron, in particular, is fascinated by mathematics. I have found my ideal readers. I have no more excuses not to proceed with At play.

One of my goals is to introduce readers to formality, ways of thinking about mathematics, complexity, and the art of algorithm design.

These books will follow a comfortable path through some areas of math, problems, and my approaches and algorithms for solving them. They will not be 'complete' in any sense. Threads on problems, approaches, and algorithms will be interwoven - I expect that the threads on the k-Clique Exists problem will continue throughout the series.

The partial-and-frequently-changing outline of the first book in the series is:

Introduction
What I think math is all about
   The greatest human art
   Constraint and consequence
   Serendipity - the usefulness of mathematics
Graphs and cliques
   Relationships and relations
   Diagrams of relationships between two things
   Graphs
   Graph theory
   Cliques
   Generalization and abstraction
   Interpretations and models
   Subgraphs
   Induced subgraphs
   Representations of graphs
      Diagram
      Adjacency matrix
      List of adjacency sets
      List of augmented adjacency sets
      Set of maximal cliques
   Maximal cliques and maximum cliques
   Some important questions about cliques
      What are all of the maximal cliques of a graph?
      What are all of the maximum cliques of a graph?
      What is the size of a maximum clique of a graph?
      Does a k-clique exist in a graph and, if so, what is an example?
Algorithms
   Recipes for doing things
Maximalizing cliques
   The maximalization of cliques
   Discovering a largest maximalization of a clique
The complexity of an algorithm
Constructing cliques using maximal matchings
The split approach
   Finding all of the maximal cliques of a graph using the split approach
   Finding all of the maximum cliques of a graph using the split approach
   Determining whether a k-clique exists in a graph using the split approach
Reduction
The k-reduction of graphs
Incremental solution
The Participants::Intersection approach
Graph isomorphism
Determining whether two graphs are isomorphic
Universal traversal sequences
Ramsey theory with respect to cliques in graphs

The first book of this series should be completed in 2011. They are intended for anyone from motivated teenagers through researchers in algorithms and relevant areas of mathematics, such as graph theory. Cameron and Gillian, their parents, and some other friends will be invited to read drafts as they become available.

Topics for future books in the series include:

• Clustering
• complexity of a problem
• patchwork coverage of a problem by algorithms
• Adaptive Market pattern
• metaphor
• search rather than research

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