At Play - My new writing project on math and algorithms

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.

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.

Some of the topics and problems that I'll start discussing in the first book are:

• graphs - directed and undirected
• strict graphs - undirected and without loop edges or multi-edges
• cliques
• maximal cliques
• clique maximalization
• subgraphs and induced subgraphs
• finding all of the maximum cliques of a graph
• finding all of the maximal cliques of a graph
• determining whether a k-clique (clique of size k) exists in a graph
• k-reduction of graphs
• clustering
• graph isomorphism
• determining whether two graphs are isomorphic
• Ramsey theory with respect to cliques in graphs
• universal traversal sequences

One of my goals is to introduce readers to formality, ways of thinking about mathematics, complexity, and the art of algorithm design. Some additional topics that I'll start discussing in the first book include:

• abstraction
• relationships and relations
• interpretations and models
• constraint and consequence
• reduction
• incremental solution
• complexity of an algorithm
• complexity of a problem
• patchwork coverage of a problem by algorithms
• Adaptive Market pattern
• metaphor
• search rather than research

The first book of this series should be completed in early 2010. 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.

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