Scientific American Editors - Martin Gardner: The Magic and Mystery of Numbers

Martin Gardner

The Magic and Mystery of Numbers

From the Editors of Scientific American

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Martin Gardner on Numbers

Mathematicians apply the word 'number' to hundreds of strange abstract beasts that are far removed from counting. Martin Gardner

Not many writers can attract a readership so broad it encompasses scientists, students, artists, writers, housewives, postmen, magicians, skeptics, philosophers and ad men. Especially when they write about math. But Martin Gardner, who would have celebrated his 100th birthday in October this year, did just thatnearly every monthfor 26 years via his Mathematical Games column in Scientific American magazine. His columns, which ran between 1957 and 1983, introduced more people to more math than perhaps any textbook, treatise or classical tome. According to Colm Mulcahy, a professor of mathematics at Spelman College in Atlanta, Georgia, Gardner was the best friend mathematics ever had.

Remarkably, Gardner was not a professional mathematician and had virtually no formal math training after high school. He did, however, possess an extraordinary intellect, along with extraordinarily diverse interests. He loved puzzles, which his father, a geologist, introduced to him as a teen, and he was fascinated by magic and its overlap with math. His first Scientific American article described hexaflexagons, flat, two-sided paper constructions (related to one-sided Mobius strips) that appear to conjure extra faces out of thin air when flipped in a certain way. And many of his columns presented popular puzzles or card tricks made less mystifying once explained in numbers.

In this eBook, Martin Gardner: The Magic and Mystery of Numbers , we strove to create a new slice through this wealth of materialbecause as any Gardner fan knows, his mathemagical tricks and games have been collected in many forms, including the six-book Scientific American series he edited himself. Regardless of the vehicle, he worked playful references and challenges into almost everything he ever wrote; and this sampling is no exception. Here, we focus on all flavors of number, from common integers and negative numbers to figurate numbers and the exotic random number, Omega, which can be described but not computed.

Some of these columns are less well known than, say, his writings about flexagons, but they are no less fun. In true Gardner fashion, they leap from magic and gamesas well as art, music and literatureto flashes of deep mathematical insight. Lattice integers become a billiards challenge and surreal numbers spawn a host of related games. The abracadabric number e, quoting French entomologist Jean-Henri Fabre, leads to spiders webs and compounded interest. The binary Gray code inspires a poem and cracks the classic Chinese Rings puzzle. Negabinary numbers, used to perform calculations on a chessboard, unveil the etymology of such words as Exchequer and bank. And binary numbers unlock mind-reading tricks and the Tower of Hanoi.

In every case, what appears here are the original columns, more or less as they first appeared. Even so, they remain current and include several conundrums still unsolvedsuch as whether the sum of irrational numbers pi and e is itself irrational. Almost every column also offers up problems for readers to solve and test their understandingalong with the answers for anyone easily frustrated. We hope that they will prove as inspirational to readers now as they did to earlier audiences.

Kristin Ozelli
Book Editor

Seeing the Integers through the Trees

The simplest of all lattices in aplanetaking the word latticein its crystallographic senseisan array of points in square formation.This is often called the lattice of integers, because if we think of the planeas a Cartesian coordinate system, thelattice is merely the set of all points onthe plane whose x and y coordinatesare integers. The illustration below shows a nite portion ofthis set: the 400 points whose coordinates range from 0 to 20.

Think of the 0,0 point as the southwest corner of a square orchard, fencedon its south and west sides but innitein its extension to the north and east.At each lattice point is a tree. If youstand at 0,0 and peer into the orchard,some trees will be visible and otherswill be hidden behind closer trees.Here, of course, our analogy breaksdown, because the trees must be takenas points and we consider any tree visible to one eye at 0,0 if a straight linefrom that point to the tree does not passthrough another point. The colored dotsmark all lattice points visible from 0,0;the unmarked grid intersections represent points that are not visible.

If we identify each point with afraction formed by placing the points y coordinate over its x coordinate, manyinteresting properties of the lattice(properties rst called to my attentionby Robert B. Ely of Philadelphia) begin to emerge. For example, each visible point is a fraction whose numeratorand denominator are coprime; that is,they have no common factor other than1 and therefore cannot be reduced to asimpler form. Each invisible point is afraction that can be simplifiedand eachsimplication corresponds to a point onthe line connecting the fraction with0,0. Consider the point 6/9 ( y = 6, x =9). It is not visible from 0,0 because itcan be simplied to 2/3. Place astraightedge so that it joins 0,0 and 6/9and you will see that the visibility of6/9 is blocked by the point at 2/3. Allpoints along the diagonals that extendup and to the right from 0/1 and 1/0are visible because no fraction whosenumerator and denominator differ by1 can be simplied.

Note that many of the diagonalsrunning the other wayfrom upper leftto lower rightconsist entirely of visible points except for their ends. Allthese diagonals, Ely points out, cutthe coordinate axes at prime numbers.Every visible point along such a diagonal is a fraction formed by two numbers that sum to the prime indicated bythe diagonals ends. Two numbers thatsum to a prime obviously must be co-prime (if they had a common factor,then that factor would also evenly divide the sum), so such fractions cannotbe simplied. Vertical and horizontallines that cut an axis at a prime get progressively denser with visible latticespots as the primes get larger, becausesuch lines have invisible lattice pointsonly where the other coordinate is amultiple of the prime.