Packing spheres, Reversi, braids, polyominoes, board games, and the puzzles of Lewis Carroll. These and other mathematical diversions return to readers in this new book, featuring updates to all the chapters, including new game variations, mathematical proofs, and other developments and discoveries. Read about Knuth’s Word Ladders program and the latest developments in the digits of pi. Once again these timeless puzzles will charm readers while demonstrating principles of logic, probability, geometry, and other fields of mathematics.

### Table of Contents

Acknowledgments

Introduction

1. The Binary System

2. Group Theory and Braids

3. Eight Problems

4. The Games and Puzzles of Lewis Carroll

5. Paper Cutting

6. Board Games

7. Packing Spheres

8. The Transcendental Number π

9. Victor Eigen: Mathemagician

10. The Four-Color Map Theorem

11. Mr. Apollinax Visits New York

12. Nine Problems

13. Polyominoes and Fault-Free Rectangles

14. Euler’s Spoilers: The Discovery of an Order-10 Graeco-Latin Square

15. The Ellipse

16. The 24 Color Squares and the 30 Color Cubes

17. H.S.M. Coxeter

18. Bridg-it and Other Games

19. Nine More Problems

20. The Calculus of Finite Differences

Index

### Excerpt: Ch. 14 Euler's Spoilers: The Discovery of an Order-10 Graeco-Latin Square (p. 175)

The history of mathematics is filled with shrewd conjectures – intuitive guesses by men of great mathematical insight – that often wait for centuries before they are proved or disproved. When this finally happens, it is a mathematical event of first magnitude. Not one but two such events were announced in April 1959 at the annual meeting of the American Mathematical Society. We need not be concerned with one of them (a proof of a conjecture in advanced group theory), but the other, a disproof of a famous guess by the great Swiss mathematician Leonhard Euler (pronounced "oiler"), is related to many classical problems in recreational mathematics. Euler had expressed his conviction that Graeco-Latin squares of certain orders could not exist. Three mathematicians (E. T. Parker, of Remington Rand Univac, a division of the Sperry rand Corporation, and R. C. Bose and S. S. Shrikhande, of the University of North Carolina) completely demolished Euler's conjecture. They found methods for constructing a infinite number of squares of the type that experts, following Euler, for 177 years had believed to be impossible.

The three mathematicians, dubbed "Euler's spoilers" by their colleagues, have written a brief account of their discovery. The following quotations from this account are interspersed with comments of my own to clarify some of the concepts or to summarize the more technical passages.

"In the last years of his life Leonhard Euler (1707-1783) wrote a lengthy memoir on a new species of magic square: *Recherches sur une nouvelle espèce de quarres magiques*. Today these cosntructions are called Latin squares after Euler's practice of labeling their cells with ordinary latin letters (as distinct from Greek Letters)."

### About the Author

For 25 of his 94 years, **Martin Gardner** wrote “Mathematical Games and Recreations,” a monthly column for *Scientific American* magazine. These columns have inspired hundreds of thousands of readers to delve more deeply into the large world of mathematics. He has also made significant contributions to magic, philosophy, children’s literature, and the debunking of pseudoscience. He has produced more than 60 books, including many best sellers, most of which are still in print. His *Annotated Alice* has sold more than one million copies.

### More Books in the MGL Series

Hexaflexagons, Probability Paradoxes, and the Tower of Hanoi

Origami, Eleusis, and the Soma Cube

Sphere Packing, Lewis Carroll, and Reversi

Knots and Borromean Rings, Rep-Tiles, and Eight Queens