The unity of combinatorics Combinatorics or the art and science of counting is a vibrant and active area of pure mathematical research with many applications. The Unity of Combinatorics succeeds in showing that the many facets of combinatorics are not merely isolated instances of clever tricks but that they have numerous connections and threads weaving them together to form a beautifully patterned tapestry of ideas. Topics include combinatorial designs combinatorial games matroids difference sets Fibonacci numbers finite geometries Pascals triangle Penrose tilings error correcting codes and many others. Anyone with an interest in mathematics professional or recreational will be sure to find this book both enlightening and enjoyable. Few mathematicians have been as active in this area as Richard Guy now in his eighth decade of mathematical productivity. Guy is the author of over 300 papers and twelve books in geometry number theory graph theory and combinatorics. In addition to being a life long number theorist and combinatorialist Guys co author Ezra Brown is a multi award winning expository writer. Together Guy and Brown have produced a book that in the spirit of the founding words of the Carus book series is accessible not only to mathematicians but to scientific workers and others with a modest mathematical background. . Cover Title page Copyright Contents Preface Credits and Permissions Introduction Chapter 1. Blocks sequences bow ties and worms 1.1. Langford sequences 1.2. Partitioning sets of integers 1.3. Penrose tilings Chapter 2. Combinatorial games 2.1. Wythoffs game 2.2. Combinatorial games: rules and examples 2.3. Developing strategies: * P positions and N positions 2.4. Nim nimbers * and the Sprague Grundy Theorem 2.5. Nim arithmetic and Nim algebra Chapter 3. Fibonacci Pascal and Catalan 3.1. Fibonacci numbers 3.2. The triangle of Pingala Al Karaji * Yang Hui and Pascal 3.3. The Catalan numbers and * the central column of Pascals triangle Chapter 4. Catwalks Sandsteps and Pascal pyramids Chapter 5. Unique rook circuits Appendix Chapter 6. Sums colorings squared squares and packings 6.1. Triples satisfying + = 6.2. Coil diagrams * and the Ringel Youngs Theorem 6.3. Squaring the square 6.4. Eulers polyhedral formula 6.5. Packings and coverings * of the complete graph 6.6. Steiner triple systems Chapter 7. Difference sets and combinatorial designs 7.1. Difference sets 7.2. Multipliers 7.3. Difference sets de Bruijn cycles * and de Bruijn graphs 7.4. Block designs Chapter 8. Geometric connections 8.1. A quick tour of projective geometry 8.2. Finite projective geometries * and Singer designs 8.3. Examples: =2 =2 and 3 8.4. Affine planes and magic squares 8.5. Heawoods map on the torus revisited 8.6. Ù and Nim 8.7. The automorphism group * of the Fano plane Chapter 9. The groups (2 7) and (3 2) and why they are isomorphic 9.1. The group (3 2) 9.2. The group (2 7) 9.3. Constructing an isomorphism of (2 7) onto (3 2).1131