Martin Gardner! He might as well have told me that his great grandpa was God. No doubt about it, Martin Gardner, creator of the witty and mind-bending “Mathematical Games” column that ran for 24 years in Scientific American, could be called the God of recreational math. And Gardner was more than that. He wrote more than 70 books on subjects as diverse as philosophy, magic and literature — The Annotated Alice, his definitive guide to Lewis Carroll’s classic, was perhaps his best selling title. He was also a leading debunker of pseudoscience: after retiring from Sci Am, he sicked his penetrating logical powers on purveyors of quackery, ESP, UFOs and the like in a column called “Notes of a Fringe Watcher,” published for 19 years in The Skeptical Inquirer.

Poet W.H. Auden, sci fi author Arthur C.Clarke, evolutionary biologist Stephen Jay Gould and astronomer Carl Sagan were among his many admirers. Vladamir Nabokov named him in a novel. Astrophysicists named an asteroid after him.

Martin Gardner died last month at the age of 95. He had continued to write and publish until the last months of his life. His last article, on the pseudoscience of Oprah Winfrey, was published in March. But he will always be remembered most fondly as bringing math to millions. Of many tributes I have read, my favorite is from mathematician Ronald Graham: “He has turned thousands of children into mathematicians, and thousands of mathematicians into children.”

It doesn’t get any better than that.

When I read some of his obits, looking for impressive biographical material, I was surprised to find that Martin Gardner, despite his godly status, struggled to keep one step ahead of his readers. When Scientific American asked him to write a column on mathematical games, he hit the secondhand bookshops to find books about math puzzles. This became his m.o. for years as he attempted to meet his monthly deadline. “The number of puzzles I’ve invented you could count on your fingers,” he told The New York Times.

Can you believe it? Martin Gardner, deity, scrambling to come up with the next mathematical game for his readers, and not always as original as we assumed. It reminds me of myself with this blog! Actually, it reminds me of myself with my books, and probably many other non-fiction authors with theirs. Students, teachers and librarians often seem to think we have a magical facility for turning the facts of the world into mellifluous, riveting prose, but actually we’re just folks who get out there to do research, write and rewrite until we’re blue in the face, and finally come up with something we’re not too embarrassed to send to a waiting readership (and reviewership) — then we hold our collective breath in the hope that someone will like it. Martin Gardner, who never took a college class in mathematics (he graduated in philosophy from the University of Chicago), wrote of the research for his Sci Am columns, “It took me so long to understand what I was writing about, that I knew how to write about it so most readers would understand it.” He took complex mathematical concepts and turned them into puzzles, explaining them clearly in playful, witty, inviting ways. According to Douglas Martin’s obituary in The New York Times, Gardner said his talent was asking good questions and transmitting the answers clearly and crisply.

Isn’t this the talent that all non-fiction authors strive to develop? For those as successful as Martin Gardner, it expresses itself effortlessly and abundantly. The rest of us it keep plugging away at it, as if it were a Martin Gardner puzzle.

Last week, I had the pleasure of working with Linda Sue Park and Ed Young at the American Embassy School of New Delhi, India. We were the featured authors at AES’s annual Authors’ Week. During one of our many dinners together, Linda Sue and I talked about the importance of reading children’s books as a prerequisite to writing children’s books. Linda Sue is a Newbury Award-winning novelist and picture book author. Although she is essentially a fiction writer, the crafts of writing fiction and non-fiction probably have more in common than they have differences, and the need for reading is surely a commonality.

In case you missed it, March 14th was an important international holiday. Every year, math enthusiasts worldwide celebrate the date as Pi Day. March 14th. 3/14. 3.14. Pi. Get it? If you’d like a higher degree of accuracy, you can celebrate Pi Minute at 1:59 on that date (as in 3.14159). Or why not Pi Second at 26 seconds into the Pi Minute (3.1415926)?

“It’s crazy! It’s irrational!” crows the website of the Exploratorium, San Francisco’s famously quirky hands-on science museum. The Exploratorium invented the holiday twenty-one years ago. In a delightful coincidence, Pi Day coincides with Albert Einstein’s birthday. Exploratorium revelers circumambulate the “Pi Shrine” 3.14 times while singing Happy Birthday to Albert.

Pi Day celebrations have spread to schools. Just over a year ago, I visited Singapore American School to give a week’s worth of presentations and I found parent volunteers serving pie to appreciative students whose math teachers were trying to sweeten their understanding of the world’s most famous irrational number. Just as pi is endless, so is the list of activities, from memory challenges and problem solving to finding how pi is connected to hat size … and writing a new form of poetry called “pi-ku,” which uses a 3-1-4 syllable pattern instead of haiku’s 5-7-5.

It is indeed the mysteriousness of pi that makes it so fascinating. For 3,500 years, according to David Blatner, author of The Joy of Pi, pi-lovers have tried to solve the “puzzle of pi” — calculating the exact ratio of a circle’s circumference to its diameter. But there is no such thing as “exact.” No matter how successful, pi can only be estimated.

A refresher course for the pi-challenged: The 16th letter of the Greek alphabet, ? or “pi,” is used to represent the number you get when you divide a circle’s circumference (the distance around) by its diameter (distance across, through the center). Try it on any circle with a ruler and string and you’ll get something a little over 3 1/8 or approximately 22/7 (some have therefore proposed the 22nd of July for Pi Day). Measured with a little more precision, the ratio comes out to 3.14. But don’t stop there. Pi is an irrational number, meaning that, expressed as a decimal, its digits go on forever without a repeating pattern. Hence the obsession of some with memorizing pi to 100, even 1,000 places. As a Pi Day gift from 5th graders at a school I visited this year on March 15th, I received a sheet of paper with pi written out to 10,000 digits. In 2002, a computer scientist found 1.24 trillion digits. Never mind that astrophysicists calculating the size of galaxies don’t seem to need an accuracy of pi any greater than 10 to 15 digits. Playing with pi offers endless hours of good, clean mathematical fun. So what if it’s irrational.

Happy (belated) Pi Day, everybody!

]]> http://davidschwartz.com/blog/happy-pi-day/feed 0 http://davidschwartz.com/blog/if-they-had-a-million-dollars http://davidschwartz.com/blog/if-they-had-a-million-dollars#comments Tue, 23 Feb 2010 01:23:31 +0000 David http://davidschwartz.com/blog/?p=433 Remember the hit song by the Barenaked Ladies, a Canadian alternative rock band (whose members are neither ladies nor naked — it least in public)? Here’s how it starts:

If I had a million dollar

If I had a million dollars

Well I’d buy you a house

I would buy you a house

I’d buy you furniture for your house…

Hate to sound like a million dollar spoilsport, but I sure wouldn’t pay a million for those lyrics. Not even a hundred. Still, the tune is darn catchy, and the refrain is a bit touching:

If I had a million dollars,

I’d buy your love.

Whether or not that particular commodity can be purchased at any price will not be the subject of this post, although you may wish to pursue it elsewhere.

Since the 1989 publication of my second book, If You Made a Million, I have seen countless examples of student writing that begin with the prompt, “If I had a million dollars…” They fall into three categories of roughly equal size. I’ll call the them “fulfillment,” “greed” and “charity.”

By “fulfillment” I mean the meeting of life’s needs and reasonable desires to improve one’s life. (OK, I admit I’ve made myself the sole judge of “reasonable” but I think you’ll sign on with me.) Those children whose million dollar dreams fall in this category tend to be realistic. With a little financial boost, they could probably buy the things they’re wishing for. And so we have six-year old Alexandra who writes in Alexandra’s Million Dollar Book (with a dedication page, incidentally, honoring one David Schwartz):

If I had a million dollars I would ask my Mom, “Can I have a book?”

If I had a million dollars I would ask my Dad, “Can I have a pet?”

I would guess that Alexandra’s life would be improved by the arrival of a pet and a book. If I could be the genie in the oil lamp, I would grant Alexandra her two modest wishes. Other children have wished for clothing, toys, trips, tickets to concerts or sporting events, their favorite foods, even — sadly —heat for their homes. Some of the cravings enter the realm of the luxurious — a swimming pool, for instance, in Opelika, Alabama. But considering the summertime heat and humidity of Alabama, I would say the swimming pool could help improve the yearner’s life (or at least comfort).

Eventually we cross the fuzzy line that separates fulfillment from greed. Here we have children gunning not for just one toy but for thousands or millions of them — often duplicates of the same thing. “I would buy a million Nintendos” or “a million Wiis” or whatever happens to be the hot gizmo du jour. But that’s only the beginning of million-dollar greed. I’ve seen kids spend their imaginary million on items that a billion dollars couldn’t buy: “I’d buy Disney World.” Not content with a such a meager piece of real estate, one spender revealed a vaster aspiration: “I’d buy Florida!” Even that was outdone by the child who boasted, “I would buy the World!”

But to me, the big winner here is philanthropy. I’m pleased to know that about a third of the authors spent their fictional money to improve the lives of others. Some took on lofty, if vague, goals like “I would buy food for the poor” while others were more specific: “I would buy a car for my brother so he can get to work.” Or, “I would give my money to my nice teacher, Mrs. Swartzel, so she can buy a golden necklace, golden ring, a golden castle. I like my teacher.” (Appropriately enough, this magnanimous student is named Buddy.) Some gave to organizations (a pie graph of “How I Spent my Million” showed sizeable slices of the pie served to worthy causes including the Leukemia Society and the Adopt a Giraffe Society). My all-time favorite came from a girl named Margarita: “I would shar my money with our teachers, the prin and the children.” (I assume the letters “cipal” were meant to come after “prin.”)

With a million in spending money, some combine greed with philanthropy. One standout example came from the writer who wanted to buy a big dinner for everyone in his school and then use any money left over to buy himself a horse, twelve dogs and a Ferrari.

Whether motivated by self-improvement, greed or philanthropy, this kind of imagining does motivate kids’ writing, but I hope their written words are accompanied verbal discussions. If I were the teacher, I would have as many questions as the kids have ways to satisfy their spending lust. For starters: “Could you really buy a million computer games for a million dollars? And why would you want a million of them anyway?”

It might be worth pointing out that many items on the kids’ wish lists, such as Disney World, would knock you back more than a million. And even with an unlimited budget, not everything in the world is for sale. The State of Florida would be on that list (though some political observers might disagree!).

Of course with a little guidance, these exercises can be mathematical as well as literary. Like something out of Math Curse, the math problems should flow like liquid dollars from the spending pumps of these students. Every time I read “If I had a million dollars, I would buy a million X,” I wonder if anyone asked the child how much each X would cost if you could buy a million for a million dollars. If they cannot come to see the unit cost as $1, an important prospective math lesson has been missed.

One 4th grade class used a million-dollar spending spree as the basis of a subtraction activity in which each child was issued a checkbook with a balance of $1,000,000. Finding real prices of items advertised in magazines and newspapers (no houses allowed and no more than one car), they had to spend their way down to $0, showing the math each step of the way. The kids were surprised to see how long it took them to dump that much money.

If only it worked that way in real life.

Before I describe a few of my faves, I must point out, as I did at length in my INK post of May 28, 2008, that many educators have inspired intermediate grade, middle school and even high school students with picture books, using them as age-appropriate teaching tools. I believe my most successful mathematical picture books are those that can be used on many levels. In fact, when asked the target age for How Much is a Million?, I often say, “Preschool through high school.” Actually, it’s not true. I should say. “Preschool through college,” but that answer might sound overly smug. (I have twice met college professors who use my book when teaching about Avogadro’s number, a behemoth number critically important in understanding quantitative chemistry.) That said, I will mention a few books that probably wouldn’t make it into the 2nd grade math classroom but should be a staple of math classrooms or libraries serving upper elementary, middle school and high school students.

In The Number Devil by German author Hans Magnus Enzensberger, a middle school-age boy named Robert dreams of travels through the world of mathematics under the tutelage of an impish devil whom he at first finds annoying but gradually comes to enjoy and admire. At the start of the book, Robert is a mediocre and indifferent math student — no surprise, considering that his ho-hum teacher at school gives the students mathematical busywork without the least bit of mathematical inspiration. (His uncomprehending mother is no better: she believes a son who voluntarily speaks of mathematical concepts must be ill!) But Robert and the devil are on an irrepressible romp and together they challenge each other while developing plethora of mathematical concepts and meeting a pantheon of famous mathematicians. All names are whimsically disguised — Leonhard Euler becomes “Owl” (Eule in German translates to “owl” in English) and roots (as in square roots, cube roots, etc.) are called “rutabagas” (which are literal roots) — just two of many such examples. Figuring out the conventional words for the concepts at hand just adds to the devilish fun of this 1997 book which is well on its way to becoming a classic.

Flatland is not on its way to classic status: it’s already there. Like The Number Devil, it incorporates dreams but the concepts are geometrical rather than numerical. This novella, written in 1884 by English schoolmaster Edwin A. Abbott, is a pointed satire of closed-minded, hierarchical Victorian society but it is also, as Isaac Asimov put it, “The best introduction one can find into the manner of perceiving dimensions.” The story has been developed into several short films and a 2007 feature film; there have been TV episodes, a role-playing game and sequels by other authors.

The narrator of the original Flatland is a square named – ready? — A. Square. He lives in a two-dimensional world, Flatland, but dreams of visiting a one-dimensional world, Lineland, where he tries to convince the ignorant monarch of a second dimension but fails to open the ruler’s eyes to a universe beyond his familiar straight line domain. He also goes to one-dimensional Pointland inhabited by a monarch who is the sole inhabitant and, in fact, is the entire universe in and of himself! (He perceives any attempts at communication from the outside world to be his own thoughts.) Later, our narrator is visited be a three-dimensional sphere, which he finds incomprehensible until he accompanies Sphere to Spaceland. From there, the two discover that the leaders of Flatland know about Spaceland and a third dimension, but they prohibit their subjects from acknowledging it, under penalty of death or imprisonment. Square tries to convince Sphere of the possibility of fourth and fifth dimensions, but the notion is soundly rejected and Square is sent home in disgrace. It’s not a happy story but fascinating and uplifting in its own way, and, like the subject matter, the book has many dimensions.

Mathematics: A Human Endeavor by Harold R. Jacobs is a wide-ranging mathematics textbook so deliciously fun and fascinating that kids actually want to read it. Even more astonishing, their parents and teachers can’t put it down either. As a college senior, I discovered its first edition (it’s now it its third) while volunteering as an enrichment provider to a select group of very sharp fifth graders, and I was hooked. Jacobs covers the breadth of high school mathematics with an emphasis on the beauty and power and real-life relevance of each subject. Aptly-chosen cartoons made me and my students laugh outloud, and we immediately saw that the cartoons weren’t included just for the guffaws. “An improvement over the square wheel,” pronounces B.C in the caption of one drawing as he flaunts his great invention: a triangular wheel. “It eliminates one bump!”Jacobs shows how B.C.’s thinking is dead wrong, and in so doing, he develops principles of polygons and the mathematical concept of limit: as the number of sides grows and approaches infinity, the polygon approaches a circle — the “wheel” with no bumps (road surface willing!).

In lucid prose, the subjects range from algebra and geometry to probability, topology, statistics and more. Jacobs often shows how the math being taught is found not only in books but in life. This book has become a staple among alternative high schools and homeschoolers, but I find it so readable that it doesn’t have to be thought of as a textbook at all. Just the same, a teacher’s guide is available.

A major step down from the literary giants I’ve just described is my own picture book targeted at upper-elementary and middle school age children. G Is for Googol: A Math Alphabet Book is a potpourri of topics from “A is for Abacus” (proficient abacus users outcompete calculator users every time) to “Z is for Zillion” (not actually number, which provides an excuse to talk about what makes a number a number). Math-loving kids of all ages have found respite in its pages, and math teachers sometimes read from it to provide literary snacks on special days. It has spawned many a student-created alphabet-book as class projects. A 6th grade class in Colorado called theirs An Algabet Anthology by “The Awesome Accelerated Academics.” At first I thought the title had a typo but I later realized that “algabet” is just a play on “algebra.” Sixth grade humor. The word play continues to the final page:

THE END

We Hope This Book

Arithmetickled You Pink

This sorry state of affairs is not limited to the United States. I am just back from speaking at primary (elementary) schools in Australia. I had a few opportunities to interact with children on the playground and I was pleased to notice the great variety of types of play, and how there seemed to be a niche for everyone. Some activities engaged solitary children, others occupied pairs or small groups, and a few involved large numbers. Yet when I shared my approving observations with teachers, I learned that, as in the U.S., recess is an endangered species.

Studies consistently prove its value. In one set of experiments from the mid-1990s, researchers found that school children became less and less attentive the longer recess was delayed. Another experimental study found that “fourth-graders were more on-task and less fidgety in the classroom on days when they had had recess, with hyperactive children among those who benefited the most.” An article in the New York Times in February, 2009, cited a study of 11,000 third graders showing that recess mitigates children’s behavioral problems. (Consider the common punishment for misbehaving children: “No recess!”) And a meta-analysis of over 200 studies suggest that physical activity during the school day results in more, and better, mental activity.

For all their lip-service to the necessity of drawing on research-based teaching strategies, education authorities in the U.S. and Australia (and probably many other countries) don’t seem to care much about research on play. It is interesting that, by contrast, China launched a nationwide “Sunlight Sport” campaign in 2007, requesting that every school offer one hour of sports and games daily to every student.

I read about Chinese children’s play and the Sunshine Sport campaign in a fascinating Australian journal called Play and Folklore, co-edited by Dr. June Factor of the University of Melbourne, an author and folklorist who writes playful and play-filled books for both children and adults. I met June at a reading conference in California about 15 years ago, and I had the good fortune of visiting her in Australia during my recent trip.

The most wonderful thing about many of June’s books for children is that they are actually by children: she is merely the compiler, and what she has compiled is straight from the mouths of kids, whom she and her university students have observed, recorded and interviewed in school playgrounds. The researchers collected children’s games, rhymes, sayings, chants, riddles, jokes and secret languages in abundance. In 2000, she published an entertaining and enlightening lexicon, Kidspeak: A Dictionary of Australian Children’s Words, Games and Sayings. The two children depicted on the cover have harsh words for each other: “Nicky woop” says one in a speech bubble, to which the other retorts, “Drongo!” (Translation: “Go away!” and “Jerk!”)

June’s collections for young readers have been loved since 1983 when Far Out, Brussels Sprout came out. It has since been joined by Real Keen, Baked Bean…Unreal, Banana Peel…Okey Dokey, Karaoke, and others in the Far Out! series. All offer a rich sampling of the linguistic range and complexity of Australian children’s vernacular language. “It’s children’s own literature,” says June, “handed down across many generations, sometimes across centuries. It’s a bridge across generations, common to childhood, not just contemporary childhood.” From Far Out, Brussels Sprout:

Quickly, quickly, I feel sickly.

Hasten, hasten, get the basin.

Ker plop!

Get the mop!

—————

Mary had a little lamb

She kept it in a closet.

And every time she let it out

It left a small deposit.

They’re not always the most proper ditties in the world. As a result, a decade ago June learned that she was the second-most censored author in Australian school libraries, after Judy Blume. She told me this with more than a hint of irony, considering that the censors were trying to save innocent children from their own words. “It tells you much about the power of adult prudery and unease about the human body and its functions — but, I hasten to add, not nearly as much as in the United States!”

Often, these censorship cases have been dismissed when the schools discovered how many families already owned the challenged books. But what disturbs June more than censors in the libraries is timekeepers on the playgrounds. “Increasingly, playtime is being restricted,” says Australia’s leading observer of playtime. “It’s happening in America and it’s happening here.” In the U.S., where we once feared a “red menace” from Asia, there now seems to be fear that Asian countries including China will overtake us not militarily but intellectually and economically. If it comes to pass, browbeating analysts should consider how our schools rejected the demonstrable benefits of playtime. American education authorities could have the demise of recess to blame for our fall from intellectual eminence. Some might call them drongos!

Child: What’s the biggest number in the whole wide world?

David: Do you think there is such a thing as the biggest number?

Audience: half “Yes,” half “No”

David: Will someone please tell me what you think the biggest number is.

Children, variously: billion, trillion, quadrillion, quintillion, googol, googolplex, etc.

David: Hang on. Let’s suppose you think “quintillion” is the biggest number. Then what about “quintillion-and-one”? Isn’t that bigger? And if that’s the biggest, what about “quintillion-and-two” — even bigger, right?

This usually leads to a triumphant retort about an enormous number familiar to many children (much less familiar to their parents and teachers):

Child: Googol has to be the biggest!

David: What’s a googol?

Many children know that “googol” is the name for a very large number — a one followed by a hundred zeros. This is an exciting concept. In my book G is for Googol: A Math Alphabet Book, I tell the story of how “googol” got its name from a nine-year old boy. Surely it is tempting to call googol “the biggest number,” but that’s a non-starter.

Me: If you think googol is the biggest number, then what about googol-and-one? Or two googol? Or a googol googol?

Almost inevitably, at this point someone proffers an even bigger number, “googolplex.” It is true that the word “googolplex” was coined to mean a one followed by a googol zeros. It’s way bigger than a measly googol! Googolplex may well designate the largest number named with a single word, but of course that doesn’t make it the biggest number. In a last-ditch effort to hold onto the hope that there is indeed such a thing as the largest number…

Child: Infinity! Nothing is larger than infinity!

True enough, but there is nothing as large as infinity either: infinity is not a number. It denotes endlessness. A number designates a specific amount.

So, finally we get to a consensus: There is no such thing as the largest number. Yet numbers as large as googol or googolplex continue to tantalize, and well they should. To me the most fascinating thing about googol is how incredibly enormous it actually is. Writing those hundred zeros, while tedious, would take only a minute or two, yet the amount represented is, as I stated in G is for Googol, “more than the number of hairs on the head of everyone in the world, more than the number of blades of grass on all the lawns of the world, more than the number of grains of sand on all the beaches of the world — even more than the number of atoms in the universe.”

The estimated number of atoms is a one followed by 72 zeros (ten to the 72nd power, but I can’t do exponents in this blog). Let’s suppose the astrophysicists who estimated the number of atoms are way off. For the moment, let’s imagine that the actual (though unknowable) number of atoms is a hundred times as what they claim. So it would be a one followed by 74 zeros —still way, way, way less than a googol.

The number “googol” is, in fact, useless — except as food for a hungry mathematical mind. And it is an especially nourishing numerical treat for young hungry minds. In fact, a child possessing just such a hungry young mind corrected me when I once said, “There isn’t a googol of anything, anywhere.” The boy countered, “There are more than a googol numbers. The number of numbers is infinite.” Right he was! Now I modify the statement: “There isn’t a googol of any physical object.”

I am less enthusiastic about the point that was made by sixth graders in a class that sent me a stack of letters. All had the same basic theme, reflected by this one:

Dear Mr. Schwartz,

How do you know how many hairs are on the head of every person in the world? You probably haven’t met every person in the world. Even if you have, babies are being born every minute. People are losing hair every day!

No argument there but, unfortunately, this class didn’t seem to have a good understanding of the importance (and legitimacy) of estimation.

Now, with the ascendancy of a certain multi-billion dollar online enterprise, it is necessary for me to include in any discussion of googol the following important inequality, lest there be confusion:

It is interesting to note that the item on the right was the result of a spelling error. When Larry Page and his friends were choosing a name for a start-up company, he attempted to name it after the huge number “googol.” Instead, he committed what is probably the most famous (and lucrative) misspelling in history. Regardless, there is absolutely no doubt that both “googol” and “google” are mighty big.

But. . . you knew there would be a “but”. . . are we depriving ourselves of anything worthwhile when we boil the art of research down to finding 30,000 google hits in 18 microseconds? I would maintain that we are, for several reasons, and I am going to write about one of them: browsing. Sometimes there is both pleasure and success to be found by poking around in the shelves of libraries or bookstores, just to see what we might find.A few years ago, I wrote G is for Googol: A Math Alphabet Book, a potpourri of enjoyable mathematical ideas in an ABC format. Unlike the many alphabet books written for young children, this one is directed at readers in the intermediate and middle school grades (as is its sequel, Q is for Quark: A Science Alphabet Book). So how am I going to fill 26 slots with delightful math? Many entries popped into my mind right away. “A” is going to be for “abacus” because I love the fact that proficient abacus users can calculate lengthy addition or subtraction problems faster than the fastest calculator user. “Z” is going to be for “zillion” because it’s not a number at all but people often don’t realize that, so by discussing the difference between real numbers that end in “illion” and fake ones, I can discuss the actual meaning of number. As for what’s going to come in between A and Z, I had lots of ideas but not enough to fill out the book, so. . . let’s browse the library!

And so it was that I stumbled across Maps, Tracks and the Bridges of Königsberg: A Book About Networks by Michael Holt, a 1975 picture book that was one of many in the now sadly defunct “Young Math” series from the now sadly defunct publishing company, Thomas Y. Crowell. What fun! A great “K” word — Königsberg. I had heard about the dilemma that the residents of Königsberg, Germany, had tried to solve — to see if they could walk across each of their city’s seven bridges exactly once (all had to be crossed once and none could be re-crossed). No one could figure out a way to do it, but for centuries they were taunted by the prospect that a hidden solution eluded them. Finally the mathematician Leonhard Euler developed the postulates and theorems of a new branch of mathematics now known as graph theory or network theory in order to solve the Königsberg bridge problem. His goal was to figure out how to walk the bridges, or to prove it impossible. He succeeded not only in why the seven Königsberg bridges could not be walked once time each, but under what circumstances it could or could not be done for any network of bridges. Network theory not only proved to have many applications (useful, for example, in designing networks of cables) but it also laid the foundation for another branch of mathematics, topology. For me, as a researcher and writer, the cool thing is that I wouldn’t have thought of including the bridges of Königsberg in G is for Googol if I hadn’t bumped into them in Holt’s book on the shelves of a school library during a break between two assembly programs.

So that was an example of browsing, more-or-less aimlessly, to see what I could find and how I could tie it in. But there is another, more directed, way that browsing the shelves has proved fruitful in my research. It’s when I know what I’m looking for but neither google, for all its power, nor the library catalog nor anyone or anything else can tell me where to find it. For example. . .

Recently I have been doing research for an upcoming book called Where in the Wild? Mysteries in Nature Concealed. . . and Revealed, the next book in the series that began with Where in the Wlld? Camouflaged Creatures Concealed… and Revealed. One of the entries in What in the Wild? will be about the diggings of a star-nosed mole, a small mammal whose activities leave unwelcome mounds of soil on lawns and pastures. Moles, whether star-nosed or not, are usually not loved by landowners whose property they think of as their own.

Books and websites about moles turned out to be informative but a bit dull, focused on the minutia of their biology or harsh methods of putting an end to them and their excavations. So I wondered if I could find an interesting book with a section about moles. My initial searches in google and the public library catalog turned up no such book, probably because any that existed did not have the word “mole” in the title or subtitle or as one of the subject terms entered into the Library of Congress Cataloging-in-Publication data. Still, I figured that at least one such book must exist. I would employ a more venerable method of research.

And so it was that I found myself on hands and knees in the Rockridge branch of the Oakland Public Library, exploring the bottom shelf of the 596s just to see what I could find.

Within a few minutes, I had walked my fingers to Every Creeping Thing: True Tales of Faintly Repulsive Wildlife by Richard Conniff. The title seemed promising, and sure enough the table of contents led me to a chapter called “Notes from the Underground” — about moles. And guess what: star-nosed moles have a starring role! Conniff describes this creature in a most quotable way as a mole that “looks as if it’s got a sea anemone stuck on its snout.” And from there followed all kinds of fascinating information about the critters themselves and a colorful curmudgeon (of the human variety) who pursues them at the behest of disgruntled landowners in England. Once again, browsing trumped google!

The other day, a student at Landstuhl Elementary/Middle School on the U.S. Army base in Landstuhl, Germany, asked me the secret of success in researching books, and I told him a few things I thought of on the spot, but it didn’t occur to me at the time to tell him what I am saying here. To rewrite Kenneth Grahame’s delightful line (which, as it happens, was spoken by the character Mole in Wind in the Willows), “There is nothing—absolutely nothing—half so much worth doing as messing about in boats.” In this case, Grahame’s ode to blissful aimlessness might be rewritten for researchers as “There is nothing so delightful — or fruitful — as messing about in libraries.”