The Chapel Hill News

November 28, 2004

And the winners are . . .


By Phillip Manning


In a recent column about the high school Mathematical Olympiads, I presented the following problem: “How many of the integers between 1 and 1,000, inclusive, can be expressed as the difference of the squares of two nonnegative integers?” The problem was a walk in the park for Olympians but tough going for most of us. So, I promised to publish the names of all who responded with the correct answer. Altogether, six people sent in solutions that were more or less right, and their names (and the right answer) are listed at the end of this column.

But what about those who could not come up with the answer. Can you learn enough mathematics to keep up with these young mathletes? And, even if you could, why should you bother? Let’s start with the first question.

Some writers try to educate their readers about mathematics with books that contain not a single equation. These books don’t work well. A book on mathematics without equations is like a book on photography without photos; they omit the essence of the subject. To grasp even the simplest math, you must learn to understand and manipulate equations.

One book that will teach you those things almost painlessly is Lynette Long’s “Painless Algebra” (Barron’s, $8.95). The book is a remarkably clear introduction to elementary mathematics. In three or four hours, a novice can pick up the basics of handling equations. After brushing up on algebra, you are now ready to tackle more sophisticated works, books that make mathematics fun. No, you didn’t read that wrong. Mathematics can be fun, even for those who consider themselves arithmetically challenged.

Many books can introduce you to the pleasures of math. I am especially fond of George Gamow’s “One Two Three ... Infinity.” I bought this book when I was in high school, and I still have it. The paperback edition cost $.50 back then. Dover sells it today for $10.95, which tells you something about how long I’ve been out of high school. Part I introduces the reader to the mystery of numbers, including the concept of imaginary numbers. The rest of the book is a wonderful introduction to science, with imaginative illustrations that accompany Gamow’s witty prose.

The “Mathematical Universe” (Wiley, $19.95) by William Dunham is more advanced but a good read nevertheless. Dunham takes the reader on a journey through the great proofs and problems of mathematics. He also digs into the lives of mathematical superstars, many of whom were a bit whacky. Paul Erdos, for example, was coddled as a youth and didn’t know how to put butter on bread until he was 21. But he devised an elegant proof of a difficult prime number theorem four years before he learned to spread butter. Erdos pursued mathematics to the exclusion of almost everything else. He never married, rarely held a job, and never established a permanent residence. He lived out of a suitcase, traveling the globe from one mathematical center to another, relying on friends to feed him and put him up. During his travels, Erdos thought about mathematics constantly and collaborated with his colleagues on problem after problem. When he died in 1996, the vagabond numbers man had coauthored more papers than anyone in the history of mathematics.

After reading these three books, you have a shot at answering the question posed at the beginning of this column. The problem appeared in the 1997 American Invitational Mathematics Examination, a test given to elite high school students. It was the first and easiest problem on the test, simple for mathematical Olympians and Paul Erdos but a challenge for the rest of us. I won’t go into detail, but with some mathematical know-how, one can prove that all the odd numbers and one-half of the even numbers (those evenly divisible by four) can be expressed as the difference between two squares. Thus, the answer is 750 integers. The proof is beyond mathematical beginners, but one doesn’t have to prove the answer to get the answer. You can solve it by starting small. Take the squares of 1 and 2 and subtract the two results. Continue along the same lines until you see patterns of odd and even numbers emerge then extrapolate to 1,000. This method is not pretty, but it works.

Six readers gave the right answer and most proved it. Actually, three of them came up with 748 or 749, but that was close enough for bragging rights. The winners are: Jerry Feng; Bruce Campbell, Chapel Hill; John Pardon; Sherry Main, a Mathcounts alumna from Walnut, California; Tim Ross, who placed 36th in the 1987 Mathcounts; and Rick Hashimoto, a seventh grade student from Madrid Middle School in El Monte, California. Congratulations to all.
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