Thatcher, Chandraswami and I

 

K. NATWAR SINGH 

How a future Prime Minister of Britain warmed to the godman with an Indian diplomat playing the reluctant translator

 

India House is among the better known diplomatic establishments in London. I first set eyes on the imposing building in 1952, when I was a student at Cambridge University. Thirty years later I entered India House as Deputy High Commissioner. One of my less attractive duties was to meet the unreasonable demands of visitors from India. Not all were disagreeable but many were.

Early in the summer of 1975, Mr. Chandraswamy telephones me. He was in London. The late Yashpal Kapoor had asked him to contact me, Chandraswamy invited me to meet me at his place. I said if he wished to see me, he should come to India House. This he did the next day. At the time he was in his late twenties. He was in his “Sadhu” attire. He did not speak a word of English. Now he does.

At this, our first meeting, he dropped names. After a few days he again come to see me. He invited my wife and me to have dinner with him.

The food was delicious. After dinner he said to us, “I will show you something you have never seen”. He then produced a large sheet of white paper and drew lines from top to bottom and left to right. Next he produced three strips of paper asked my wife to write a question on each strip, make a ball and place each one on a square on the chess board. My wife wrote the questions in English. He closed his eyes and went into a trance. I was, by this time getting restless. Suddenly he asked my wife to pick up any of the paper balls. She did so. Opened it. Chandraswamy then told her what the question was. He was spot on. My wife, who is an amateur astrologer, was sceptical at this stage. When Chandraswamy got the next two questions right, she was amazed and interested. I was intrigued. I could not, as a rationalist, accept mumbo-jumbo. Neither could I dismiss Chandraswamy as a complete hoax.

A few days later Y.B. Chavan, the then External Affairs Minister was on his way to the United States. I went to meet him at London’s Heathrow airport. He confirmed he knew Chandraswamy well. I also told Chavan that Chandraswamy had asked me to arrange a meeting with Lord Mountbatten and also with Mrs Thatcher. Should I arrange these meetings? To my discomfiture and surprise, Chavan sahib saw no harm in Chandraswamy meeting Lord Mountbatten or Mrs Thatcher.

I rang up Lord Mountbatten. He said he would have been glad to meet “your friend”, but he was leaving for a holiday in Northern Ireland the next day. I was quite relieved. I informed Chandraswamy. What about Mrs Thatcher?

She had been elected leader of the Conservative Party six months earlier. Doubts still assailed me about Chandraswamy meeting Margaret Thatcher, not yet the iron lady. Suppose Chandraswamy made an ass of himself. I would look a bigger ass. I sought an appointment with the Leader of the Opposition. She promptly obliged. I met her in her tiny office in the House of Commons.

Her response was, “If you think I should meet him, I shall. What does he want to see me for?” “That he will tell you himself,” I said. She agreed to see him in her House of Commons office early the next week. “Only ten minutes, Deputy High Commissioner,” she announced. I thanked her and left.

Chandraswamy was on cloud nine when I gave him the news. I cautioned him not to do or say something silly. I was putting my neck on the line for him. “Chinta mut kareay, (don’t worry”) said the sage. So, to the House of Commons the two of us proceeded. Chandraswamy was dressed in his “sadhu” kit, with a huge tilak on his forehead and a staff in his right hand. Rudraksha malas round his neck. He banged the staff on the road till I told him to stop doing so. I confess, I was feeling self conscious. Not Chandraswamy. He relished the attention he was inviting. Finally we reached Mrs Thatcher’s office. With her was her Parliamentary Private Secretary, Adam Butler, M.P. son of Rab Butler, the Conservative leader.

Introductions over, Mrs Thatcher asked, “What did you want to see me for?” Chandraswamy spoke in Hindi. I translated. “Tell her she will soon find out.” His tone was arrogantly respectful. Mrs Thatcher — “I am waiting.” The clock was ticking away. Chandraswamy was in no hurry. He asked for a large piece of paper. Went through the same routine as with my wife. He gave Mrs. Thatcher five strips of paper and requested her to write a question on each. She obliged, but with scarcely camouflaged irritation. Chandraswamy asked her to open the first paper ball. She did. He gave the text of the question in Hindi. I translated. Correct. I watched Mrs Thatcher. The irritation gave way to curiosity. Next question. Again bull’s eye. Curiosity replaced by interest. By the fourth question the future iron lady’s demeanour changed. She began to look at Chandraswamy not as a fraud, but as a holy man indeed. My body language too altered. Last question. No problem. I heaved a sigh of relief. Mrs Thatcher was now perched on the edge of the sofa. Like Oliver Twist, she asked for more. Chandraswamy was like a triumphant Guru. He took off his chappals and sat on the sofa in the lotus pose. I was appalled. Mrs Thatcher seemed to approve. She asked supplementary questions. In each case Chandraswamy’s response almost overwhelmed the future Prime Minister. She was on the verge of another supplementary, when Chandraswamy regally announced that the sun had set. No more questions. Mrs Thatcher was not put out. She enquired if she could meet him again. I was entirely unprepared for this. Very coolly, almost condescendingly he said, “On Tuesday at 2.30 p.m. at the house of Shri Natwar Singh.” I told him that he was over reaching himself by dictating the day and time without taking into account her convenience. This was not India. He was unmoved “Kunwar sahib, Anuvad kar dijiye aur phir dekhiye.” Please translate and then see. I was astounded when she asked me, “Deputy High Commissioner, where do you live?” This was not all. What followed was something out of a weird novel. Just as we were about to leave, Mr. Holy Man produced a talisman tied to a not so tidy piece of string. He then pronounced that Mrs Thatcher should tie it on her left arm when she came to my house on Tuesday. I was now on the verge of losing my temper. I said I would not translate this dehati rubbish. Mrs Thatcher intervened to know what the holy man was saying. “Mrs Thatcher, please forgive me, but Chandraswamy would like you to wear this talisman on your left arm.” She took the talisman. We were saying our goodbyes, when Chandraswamy produced his sartorial bomb. Turning to me he said “Kunwar Sahib, kindly tell Mrs Thatcher that on Tuesday she should wear a red poshak. I felt like hitting him. He was overdoing this. I firmly told him it was the height of bad manners to tell a lady what she should or should not wear. Mrs Thatcher looked a bit apprehensive at this not so mild altercation between a distraught Deputy High Commissioner and a somewhat ill-mannered holy man. Very reluctantly I said to her that the holy man would be obliged if she wore a red dress on Tuesday. I was looking down at the floor as I said this.

On Tuesday, at 2.30, Mrs Margaret Thatcher, leader of the Conservative Party, arrived at Sun House, Frognal Way, Hampstead. It was a beautiful day. She was wearing a stunning red dress. The talisman too was in its proper place.

She asked many questions but the most important related to the chances of her becoming Prime Minister. My wife was also present. Chandraswamy did not disappoint Mrs Thatcher. He prophesied that she would be Prime Minister for nine, eleven or thirteen years. Mrs Thatcher, no doubt believed that she would be Prime Minister one day. Nine, eleven, thirteen years was a bit much. Mrs Thatcher put one final question. When would she become a Prime Minister? Chandraswamy announced — in three or four years. He was proved right. She was PM for eleven years.

This narrative should have ended here. But there was an aftermath. The Commonwealth Summit was held in Lusaka, Zambia in 1979. Mrs Margaret Thatcher had by then become Prime Minister. I had been posted to Zambia in August 1977. Along with other High Commissioners I went to Lusaka airport to receive Mrs Thatcher. When she greeted me and my wife, I gently whispered “Our man proved right.” For a moment she looked flustered. She took me aside, “High Commissioner, we don’t talk about these matters.” “Of course not, prime minister, of course not,” said I.

(Extracted from K. Natwar Singh’s new book “Walking with Lions — Tales from a Diplomatic Past,”HarperCollins)

Pritish Nandy - My separated at birth twin

The pleasures of being a bore

Pritish Nandy
29 May 2012, 12:03 AM IST

I am an itinerant presence on Twitter. I am not on Facebook. I rarely trawl malls and try out new brands, new restaurants. I avoid pulp fiction and Hollywood blockbusters don't excite me overmuch. Dating a celebrity is not exactly my idea of a great evening out. And no, I don't go to Ibiza to party or Bahrain for F1. I don't even own a Blackberry or an iPad. I haven't worn a watch in years but am almost always on time. And no, I don't consider myself famous, never did.
Now doesn't this make me the perfect bore?

I write for my livelihood, paint for my pleasure. I make movies because they are fun. I work out because it makes me feel good. I yoga because it wakes me up early and allows me to watch the city come to life. I tweet when I feel like and I enjoy the response of others to what I say, even when they are not always polite. The interplay of ideas sharpens my thoughts. I walk into bookshops, sit in a corner and read. I travel a lot because it allows me to escape the ennui of routine. You can recognise me anywhere by my faded jeans and white shirt. A grey waistcoat and sneakers complete the ensemble. I never dyed my beard which greyed in my thirties. I shaved my head by accident and liked it so much I never grew my hair back.

I listen to all music, enjoy them all. From Elvis to Gangubai Hangal to Nusrat to Adele. But yes, I love music where the words touch my heart. I love Sahir and Kaifi. I re-read old classics. But I enjoy watching The Simpsons too. It bothers me when Inception tests my intelligence, and my patience. But that doesn't mean I watch Houseful 2. I would rather watch ZNMD or Kahaani. My idea of a perfect date would be in a tiny café in a place where I have never been with someone I have never met and am unlikely to ever meet again. Mystery and magic are what I seek from life, and the occasional miracle of love.

So rarely do I go to parties that people have stopped inviting me. The company of one beautiful or intelligent person excites me far more than people in the collective trying very hard to enjoy themselves. I find the world a charming place, best savoured on one's own or with someone you love. Group celebration is as unexciting to me as group sex. I find both tedious. Sex, like love, is at its best when you experience it with someone of the opposite sex, which makes me doubly boring in a world where almost everyone is bisexual or (in Samantha's memorable coinage) trysexual. I really wouldn't know what to do with a naked man. Only women exist in my sexual universe.

Even there I am deadly boring. S&M doesn't titillate me. Mozart may. I passed on drugs when I passed out of school. Alcohol makes me drowsy. And the current obsession over food I find gross. I eat little, speak less, grab the passing moment. Neither greed nor gluttony excite me. I wouldn't notice if Gordon Ramsay was in the kitchen. It's the person I am with who makes it happen. I never eat alone. The only food I miss is what I don't get. Ergo, nostalgia food. A meal I had on a steamer in Bangladesh. My mother's cooking, even though it was never great. I miss food from little known places that have shut down. I remember a city by what I ate there, usually happenstance street food.
I believe our hearts teach us how to react. A book, a film, a song may move me to tears at a special moment. On another, they could leave me untouched. That's why it's so tough being a critic. You have to carry your moment with you. Trees, dogs, cats, birds, flowers, squirrels running on the fence, the sound of laughter work any time for me, and the delight of walking through unknown streets, empty fields, unseen dreams. I love them all and wish I could pass on the memories to those I care for instead of the trinkets we gift each other and so easily forget.

Ian Stewart's top 10 popular mathematics books

Ian Stewart is an Emeritus Professor of Mathematics at Warwick University and a Fellow of the Royal Society. He has written over 80 books, mainly popular mathematics, and has won three gold medals for his work on the public understanding of science. In collaboration with Terry Pratchett and Jack Cohen he wrote the Science of Discworld series. His new book, 17 Equations That Changed the World, is published by Profile.

1. Seventeen Equations that Changed the World
2. by Ian Stewart
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"'Popular mathematics' may sound like a contradiction in terms. That's what makes the genre so important: we have to change that perception. Mathematics is the Cinderella science: undervalued, underestimated, and misunderstood. Yet it has been one of the main driving forces behind human society for at least three millennia, it powers all of today's technology, and it underpins almost every aspect of our daily lives.
"It's not really surprising that few outside the subject appreciate it, though. School mathematics is so focused on getting the right answer and passing the exam that there is seldom an opportunity to find out what it's all for. The hard core of real mathematics is extremely difficult, and it takes six or seven years to train a research mathematician after they leave school. Popular mathematics provides an entry route for non-specialists. It allows them to appreciate where mathematics came from, who created it, what it's good for, and where it's going, without getting tangled up in the technicalities. It's like listening to music instead of composing it.
"There are many ways to make real mathematics accessible. Its history reveals the subject as a human activity and gives a feel for the broad flow of ideas over the centuries. Biographies of great mathematicians tell us what it's like to work at the frontiers of human knowledge. The great problems, the ones that hit the news media when they are finally solved after centuries of effort, are always fascinating. So are the unsolved ones and the latest hot research areas. The myriad applications of mathematics, from medicine to the iPad, are an almost inexhaustible source of inspiration."

1. The Man Who Knew Infinity by Robert Kanigel

The self-taught Indian genius Srinivasa Ramanujan had a flair for strange and beautiful formulas, so unusual that mathematicians are still coming to grips with their true meaning. He was born into a poor Brahmin family in 1887 and was pursuing original research in his teens. In 1912, he was brought to work at Cambridge. He died of malnutrition and other unknown causes in 1920, leaving a rich legacy that is still not fully understood. There has never been another mathematical life story like it: absolutely riveting.

2. Gödel, Escher, Bach by Douglas Hofstadter

One of the great cult books, a very original take on the logical paradoxes associated with self-reference, such as "this statement is false". Hofstadter combines the mathematical logic of Kurt Gödel, who proved that some questions in arithmetic can never be answered, with the etchings of Maurits Escher and the music of Bach. Frequent dramatic dialogues between Lewis Carroll's characters Achilles and the Tortoise motivate key topics in a highly original manner, along with their friend Crab who invents the tortoise-chomping record player. DNA and computers get extensive treatment too.

3. The Colossal Book of Mathematics by Martin Gardner

In his long-running Mathematical Games column in Scientific American, Gardner – a journalist with no mathematical training – created the field of recreational mathematics. On the surface his columns were about puzzles and games, but they all concealed mathematical principles, some simple, some surprisingly deep. He combined a playful and clear approach to his subject with a well-developed taste for what was mathematically significant. The book consists of numerous selections from his columns, classified according to the mathematical area involved. Learn how to make a hexaflexagon and why playing Brussels sprouts is a waste of time.

4. Euclid in the Rainforest by Joseph Mazur

A thoroughly readable account of the meaning of truth in mathematics, presented through a series of quirky adventures in the Greek Islands, the jungles around the Orinoco River, and elsewhere. Examines tricky concepts like infinity, topology, and probability through tall tales and anecdotes. Three different kinds of truth are examined: formal classical logic, the role of the infinite, and inference by plausible reasoning. The story of the student who believed nothing except his calculator is an object lesson for everyone who thinks mathematics is just 'sums'.

5. Four Colours Suffice by Robin Wilson

In 1852 Francis Guthrie, a young South African mathematician, was attempting to colour the counties in a map of England. Guthrie discovered that he needed only four different colours to ensure that any two adjacent counties had different colours. After some experimentation he convinced himself that the same goes for any map whatsoever. This is the remarkable story of how mathematicians eventually proved he was right, but only with the aid of computers, bringing into question the meaning of "proof". It contains enough detail to be satisfying, but remains accessible and informative throughout.

6. What is Mathematics Really? by Reuben Hersh

The classic text What is Mathematics? by Richard Courant and Herbert Robbins focused on the subject's nuts and bolts. It answered its title question by example. Hersh takes a more philosophical view, based on his experience as a professional mathematician. The common working philosophy of most mathematicians is a kind of vague Platonism: mathematical concepts have some sort of independent existence in some ideal world. Although this is what it feels like to insiders, Hersh argues that mathematics is a collective human construct – like money or the Supreme Court. However, it is a construct constrained by its own internal logic; it's not arbitrary. You choose the concepts that interest you, but you don't get to choose how they behave.

7. Magical Mathematics by Persi Diaconis and Ron Graham

Both authors are top-rank mathematicians with years of stage performances behind them, and their speciality is mathematical magic. They show how mathematics relates to juggling and reveal the secrets behind some amazing card tricks. Here's one. The magician mails a pack of cards to anyone, asking them to shuffle it and choose a card. Then he shuffles the cards again, and mails half of them to the magician—not saying whether the chosen card is included. By return mail, the magician names the selected card. No trickery: it all depends on the mathematics of shuffles.

8. Games of Life by Karl Sigmund

Biologists' understanding of many vital features of the living world, such as sex and survival, depends on the theory of evolution. One of the basic theoretical tools here is the mathematics of game theory, in which several players compete by choosing from a list of possible strategies. The children's game of rock-paper-scissors is a good example. The book illuminates such questions as how genes spread through a population and the evolution of cooperation, by finding the best strategies for games such as cat and mouse, the battle of the sexes, and the prisoner's dilemma. On the borderline between popular science and an academic text, but eminently readable without specialist knowledge.

9. Mathenauts: Tales of Mathematical Wonder edited by Rudy Rucker

A collection of 23 science fiction short stories, each of which centres on mathematics. Two are by Martin Gardner, and many of the great writers of SF are represented: Isaac Asimov, Gregory Benford, Larry Niven, Frederik Pohl. The high point is Norman Kagan's utterly hilarious "The Mathenauts", in which only mathematicians can travel through space, because space is mathematical – and, conversely, anything mathematical can be reality. An isomorphomechanism is essential equipment. Between them, these tales cover most of the undergraduate mathematics syllabus, though not in examinable form.

10. The Mathematical Principles of Natural Philosophy by Isaac Newton

There ought to be a great classic in this top 10, and there is none greater. I've put it last because it's not popularisation in the strict sense. However, it slips in because it communicated to the world one of the very greatest ideas of all time: Nature has laws, and they can be expressed in the language of mathematics. Using nothing more complicated than Euclid's geometry, Newton developed his laws of motion and gravity, applying them to the motion of the planets and strange wobbles in the position of the Moon. He famously said that he "stood on the shoulders of giants", and so he did, but this book set the scientific world alight. As John Maynard Keyes wrote, Newton was a transitional figure of immense stature: "the last of the magicians … the last wonderchild to whom the Magi could do sincere and appropriate homage." No mathematical book has had more impact.

• © 2012 Guardian News and Media Limited or its affiliated companies. All rights reserved.

Pick a Card, Any Card

• DECEMBER 10, 2011 The Wall Street Journal

By ALEX STONE

The standard way to mix a deck of playing cards—the one used everywhere from casinos to rec rooms—is what is known as a riffle (or "dovetail") shuffle. You begin by splitting the deck into two roughly equal stacks. Then you flick the cards with your thumbs off the bottoms of the piles in alternating fashion, interleaving the two stacks.

For games like blackjack or poker to be truly fair, the order of the cards must be completely random when the game begins. Otherwise a skilled cheat can exploit the lack of randomness to gain an advantage over other players.

How many riffle shuffles does it take to adequately mix a deck of 52 playing cards?

Francesco Abrignani/Alamy

As it turns out, you have to shuffle seven times before a deck becomes truly scrambled. Not only that, the cards become mixed in a highly unusual way: The amount of randomness in the deck does not increase smoothly. The first few shuffles do little to disturb the original order, and even after six shuffles, you can still pick out distinctly non-random patches.

But right around the seventh shuffle something remarkable happens. Shuffling hits its tipping point, and the cards rapidly decay into chaos.

Magical Mathematics

By Persi Diaconis and Ron Graham
Princeton, 244 pages, $29.95

The seven-shuffles finding applies to messy, imperfect riffle shuffles. The deck might not be divided exactly in half, for instance, or the cards might be riffled together in a haphazard way. Far from undesirable, a little sloppiness is actually the key to a random shuffle.

A perfect (or "faro") shuffle, meanwhile, wherein the deck is split precisely in half and the two halves are zippered together in perfect alternation, isn't random at all. In fact, it's completely predictable. Eight perfect shuffles will return a 52-card deck to its original order, with every card cycling back to its starting position.
And this doesn't just work for 52 cards. A deck of any size will eventually return to its starting order after a finite sequence of faro shuffles, although the number of faros required isn't always eight—and doesn't increase linearly. If you have 104 cards, for instance, it takes 51 faros to restore the deck. For a thousand cards, it takes 36.

These findings are among the many fascinating results explored in "Magical Mathematics," a dazzling tour of math-based magic tricks. The authors, Persi Diaconis and Ron Graham, are distinguished mathematicians with high-powered academic pedigrees. Both are also accomplished magicians who have taught courses on mathematical magic at Harvard and Stanford.

Mr. Diaconis has an especially unusual résumé for a mathematician. In 1959, at age 14, he ran away from home to study with the great 20th-century sleight-of-hand master Dai Vernon—a man who once fooled Harry Houdini with a card trick. After spending 10 years under Vernon's tutelage, Mr. Diaconis returned home to New York and enrolled in night school, eventually earning a full ride to a Ph.D. program in mathematics at Harvard.

The book's title may strike some people as odd in its pairing of magic and math, but the two subjects share a common lineage that goes back centuries. In fact, some of the earliest recorded magic tricks were based in math. Fibonacci's 1202 manuscript "Liber Abaci," the foundation of modern arithmetic, contains a number of magic tricks, including several versions of the famous three-object divination, wherein a spectator mentally selects one of three objects and the magician correctly identifies the spectator's choice.

The earliest recorded card tricks, meanwhile, appear in a math text written around 1500 by a Tuscan friar who was close friends with Leonardo da Vinci. And one of the first magic manuals was compiled in the 17th century by Claude Gaspard Bachet de Méziriac, an early number theorist.

Player/Alamy

But mathematical magic truly came of age in the 20th century, with the growth of magic as a mainstream hobby. "In the past hundred years, a revolution has taken place," the authors write, citing the thousands of math-based magic tricks now in circulation.

In their breezy yet authoritative book, Messrs. Diaconis and Graham showcase some of the genre's best creations as well as many new ones of their own devising. Included are tricks with coins and cards (the reader will want to have a deck handy), a divination routine that employs the I Ching—the 5,000-year-old Chinese fortune-telling book—and, my personal favorite, a gambling demonstration in which the spectator shuffles a deck of cards but somehow still manages to deal himself a royal flush in spades.

This last effect exploits something known as the Gilbreath Principle, a beautiful property discovered in the 1950s by a mathematician who worked for many years at the Rand Corp. Take a deck of cards and arrange it in alternating red-black order. Now deal half of the deck facedown into a pile—thus reversing its order—and riffle shuffle the two piles together. Finally, deal the cards face up in pairs.

Each pair will contain one red and one black card (though not necessarily in alternating order). This is the Gilbreath Principle. This same idea applies to any repeating pattern of cards. If, for instance, the deck is arranged so that the cards cycle through the four suits—clubs, hearts, spades, diamonds, clubs, hearts, spades, diamonds, and so on throughout the deck—and the same procedure is executed, then every four cards dealt off the top will contain a complete set of suits. This result, combined with a few clever subtleties, is the basis of the royal-flush effect.

All the tricks in "Magical Mathematics" are of the "self-working" variety—meaning they require little or no physical skill—and while a grasp of the underlying mathematics is helpful, it is by no means a necessity. Even math-phobes will be able to astound audiences by simply following the directions and consulting the many full-color illustrations provided throughout the text.

The mixing of magic and math is more than just a means to new tricks. It has also spawned a host of major mathematical breakthroughs. "Some magic tricks use 'real mathematics' and lead to questions beyond the limits of modern mathematics," the authors write. "Sometimes, we have been able to solve the math problems."

The seven-shuffles result is one such solution. Mr. Diaconis became interested in the math of shuffling after he encountered a card trick published in the early part of the 20th century by Charles Jordan, a chicken farmer and champion puzzle solver who invented several groundbreaking card tricks. In this particular effect—called "Long Distance Mind Reading," because it could be performed through the mail—the spectator shuffles before and after picking a card, but the magician still finds his selection.

Mr. Diaconis realized that for the trick to work shuffling had to be less effective than people generally assumed. While at Harvard, he teamed up with a mathematician named David Bayer and the two undertook a theoretical analysis, building on work done at Bell Labs in the 1950s. Their landmark 1992 paper—"Trailing the Dovetail Shuffle to its Lair"—rigorously proved that anything less than seven shuffles is inadequate. Not only that, their results had implications for a wide class of "mixing" phenomena—from stirring cake batter to compounding chemicals.

Similarly, the remarkable "looping" property of perfect shuffles is a facet of group theory—a branch of abstract mathematics that deals with, among other things, symmetric structures. Group theory has applications to chemistry, biology and, most notably, physics, where it provides the mathematical framework for the Standard Model—the overarching theory of subatomic particles and forces.

There's also a deep link between the perfect shuffle and the binary number system—the universal language of modern computing. To appreciate the connection, you first have to understand that there are two ways to do a faro. You can either weave the cards together so that the top and bottom cards stay in place—this is called an "out-faro"—or you can do what is known as an in-faro, in which the top and bottom cards each move inward by one card.

Now let's say that the ace of spades is on top, and you want to move 25 cards above it, so that the ace will be 26th from the top. The sequence of faros required to bring about this arrangement can be found by writing the number 25 in binary notation, like this: 11001. For each 1, you do an in-faro, and for each 0 you perform an out-faro. In this case, you would do two in-faros (11), followed by two outs (00) and, lastly, one more in (1).

Shuffling is one example of something seemingly ordinary that subtends an elegant mathematical structure. Juggling is another. "Mathematics is often described as the science of patterns," Messrs. Diaconis and Graham (a former president of the International Jugglers' Association) write. "Juggling can be thought of as the art of controlling patterns in time and space. Both activities offer unbounded challenges."

The central challenge in the mathematical study of juggling is to figure out which sequences of throws are possible and to categorize them according to the number of balls they require and their length—or period. Toward that end, mathematicians have developed a notation, called "siteswap," that uniquely describes all possible throwing sequences.

A siteswap pattern consists of a string of numbers, each of which specifies how much time one ball—or club, or chainsaw, or banana—spends in the air. The classic three-ball cascade, for instance, is denoted 333, because each ball is aloft for the same amount of time (three beats), and the sequence repeats after every third throw.

The remarkable thing about siteswap is that it allows jugglers to devise new patterns on paper and determine whether they're juggleable with a few simple calculations, all without tossing a single ball. What's more, the average of the digits in a pattern tells you the number of objects needed to juggle it—3 in the case of 441, for example, since the average of 4, 4 and 1 is 3.

Siteswap has led to the discovery of hundreds of unknown throwing sequences, many with just three or four balls. "Once the connection has been made between juggling (sequences) and mathematics, all kinds of doors, both mathematical as well as juggling, are thrown wide open," the authors note. "Many jugglers have been working hard to master the almost unlimited number of new patterns suggested by siteswaps."

Throughout the book, Messrs. Diaconis and Graham shuttle back and forth between magic and math, probing each trick for hidden mathematical insights and developing new magic based on what they find. In the process, they encounter a number of unsolved problems, some of which have prize money attached to them. It's a fun ride, even if you don't follow the nuances of every theorem and proof, and a refreshing change from the bombastic sort of magic one typically encounters on television.

Lovers of recreational mathematics, and especially fans of the late Martin Gardner, who contributed the foreword, will find many pleasures in "Magical Mathematics." And while exposing magic secrets in a book intended for the general public may raise hackles among some old-guard magicians, exploring the math behind these tricks will, in truth, only deepen the mystery. For, as the authors remind us, sometimes the methods are as magical as the tricks themselves.
 

—Mr. Stone is the author of the forthcoming "Fooling Houdini: Magicians, Mentalists, Math Geeks, and the Hidden Powers of the Mind."