At 10 months Adam, according to his parents, suddenly said to them, "Please teach me logarithms. I understand the characteristic but I don't understand the mantissa ..."
-- Tufts University psychologist David Feldman, lecturing on child prodigies
Regal in her sari, Shakuntala Devi stood before an audience at George Washington University, her hands clasped like a singer's. The music that came out, however, was numbers.
The cube of 121? "One seven seven one five six one," she replied instantly. (She hates commas.) Then, after a moment's reflection: "That's also the fourth power of 11. There's something much more interesting to this number than meets the eye."
In fact it is the sixth power of 11, a lapse no one noticed.
Someone asked if she could handle decimals and wanted her to find the cube root of 12812.904. That, she said without hesitation, is the third power of 23.4.
Then it was days of the week. Given a date in the distant past or future, she answered "Thursday" or "Sunday" or "Tuesday" almost before the questioner had got the words out. She was always right. She played with her hotel room number, 1729: It is the sum of 12 cubed and one cubed, also the sum of 10 cubed and nine cubed.
But this was child's play for the plump, fiftyish Devi, one of the world's most celebrated calculating prodigies. She made the Guinness Book of Records a few years ago by multiplying two 13-digit numbers -- correctly, of course -- in 28 seconds.
In 1977 she made headlines all over the world when she beat the Univac computer by figuring in her head, before a rapt audience at Southern Methodist University in Dallas, the 23rd root of a 201-digit number. In 50 seconds. The machine, which had to be specially programmed for this event, took more than a minute.
She says it just comes to her, the numbers appear in her mind. She says her gift is from God and doesn't think her mind would cooperate if she performed in a communist country. She says it is terrible the way we let machines do our memory work for us.
Yet when she appeared at GWU the other day she announced it was her first official performance in the United States, though actually she had visited here in 1976 and 1952. And when asked, "What day is it today?" she wasn't sure.
"I have to be relaxed, " she says. "I try to clear my mind. I don't watch TV on the day I perform, I don't get into conversations. I can work about 90 minutes, and then I get tired ... "
Everyone is fascinated by "lightning calculators," "human computers" and idiot savants -- who, at an age when most children are barely aware they're alive, can do astonishing, almost miraculous things in one narrow field but who remain ordinary, or even subnormal, in everything else.
Many of them turn up in music and math (also chess), disciplines that resemble each other in their architectural qualities, their purity as abstract art, their freedom from the imprecisions of language or any other form of intellectual interpretation.
The musicians are the most famous: little Mozart composing symphonies in the attic in his pajamas; 2-year-old Claudio Arrau reaching up over his head to touch the piano keys and discover his calling; Menuhin and Chopin giving concerts in knee pants, and so on. Much rarer are the prodigy artists like Nadia, an autistic girl who at kindergarten age was drawing with the sophisticated skill of a graduate art student.
The gift seems to appear earliest in the math prodigies, often before they have any inkling that there is such a thing as mathematics, which is why so many of them reinvent it for themselves while musing in their highchairs.
For all the fascination of the phenomenon, remarkably little has been written on it. The Institute of Noetic Sciences held a two-day conference here recently on "The Greater Self: New Frontiers in Exceptional Abilities Research," exploring the human potential in terms of, among other things, Tibetan meditation techniques, remission of disease, business performance, child prodigies and the great calculators.
The nonprofit institute was founded 14 years ago by astronaut Edgar Mitchell to explore the nature of consciousness and the innumerable, subtle connections between mind and body. Mitchell told the audience of 1,400 that his perspective on life was changed forever by seeing the planet as a whole, a ball in space, and he urged that thinking humans learn to accept major change with equanimity.
One feature of the conference was a demonstration by Hans Eberstark, an engaging genius who has memorized pi to 11,944 places and speaks at least two dozen languages.
He had the audience call out 50 digits, which were copied on a screen behind him. When he recited them, going slowly and methodically, he left out a chunk of 10 but soon recovered and got them right. It was clear he was using a memory system, and later he explained that he translates each digit into a sound, then works these homemade syllables into a private jargon.
Compared to Devi and some prodigies of the past, Eberstark was rather reassuring in his hesitations and false starts ... though it should be emphasized that he wasn't exactly pushing himself in his brief and chatty appearance.
He was introduced by his friend Steven B. Smith, himself a numbers whiz, who has written probably the best book ever on "The Great Mental Calculators." For those who are simply flabbergasted by such math gymnastics, who start muttering about deviant brain cells and the supernatural, Smith's book is a revelation.
For instance, about that feat of Devi's in beating the computer, he has this to say:
"The computer apparently did not, as did Devi, extract the 23rd root of a 201-digit number where the root was known to be an integer, but rather raised a nine-digit number to the 23rd power. The problems are altogether different. If the computer had been given the same 201-digit number and programmed to use methods similar to Devi's, it would have given the answer virtually instantaneously, while no one could conceivably raise an arbitrary nine-digit number to the 23rd power."
Incidentally, Smith finds Devi's 28-second multiplication of two 13-digit numbers frankly "unbelievable" because "it is so far superior to anything previously reported."
But can these dazzling gifts be reduced to merely a matter of method and technique? Many of the famous calculators in Smith's book were illiterate, knew nothing about arithmetic (at least when they started). Some learned to multiply by arranging pebbles in rectangles. And when you consider the speed -- Wim Klein of Holland extracted the 73rd root of a 500-digit number in under three minutes -- it's hard to believe there is time for any technique.
And what about those people who can tell you, just like that, the day of the week for any date within thousands of years, yet who spend their lives in mental institutions, diagnosed as retarded, and couldn't possibly have memorized some formula?
The fact is, numbers prodigies cover a vast panorama of talent, ranging from John and Michael, the much-televised twins with IQs of 60 who could quote, offhand, prime numbers 20 digits long, to mathematicians John von Neumann and Karl Gauss, scientist Andre' Ampe`re and the wonderful Alexander Aitken, linguist, composer, violinist, poet and an instant calculator of the first order.
And nearly all of these people do have methods, though some wouldn't use the word. Rather, they would say they are in love with numbers, they play with them day and night, they delight in the myriad ways numbers relate to each other, create harmonies in the mind.
Wim Klein used to say, "Numbers are friends for me, more or less. It doesn't mean the same for you, does it, 3,844? For you it's just a three and an eight and a four and a four. But I say, 'Hi, 62 squared.' "
Salo Finkelstein thought 214 "beautiful," was especially fond of 8337, hated zero. Shyam Marathe, flying over the Grand Canyon, was inspired to revel in the vastness of the 20th power of nine. Eberstark sees "the sinister 64 or the arrogant, smug, self-satisfied 36 ... the fatherly, reliable (if somewhat stodgy) 76."
"Perhaps this is because many calculating prodigies were children," Eberstark writes in an introduction to Smith's book; "perhaps because numbers, like puppy dogs, befriend those who want to play with them; probably because, as we shall see, numbers are ideal toys."
The sheer act of counting can become almost a tic with some of these people. The English peasant Jedediah Buxton, a prodigious counter who once calculated the number of inch-long hairs in a cubic mile, was taken in 1754 to see David Garrick in "Richard III," and he spent the time mainly counting the words the actor uttered.
Many calculators, as part of their constant fooling around with numbers, habitually factor any large figure they see. Thus, at a moment's notice they can dismantle a number like a toy, into more workable bits. Some memorize the multiplication tables up to 100 and beyond. A few memorize logarithm tables.
Smith gives a glimpse of Finkelstein's mind at work:
"Problem: reduce 6,328 to the sum of four squares. Thought that 71 =5,041. Thought of subtracting it; didn't like it, so didn't. Thought 72 . Doesn't know it. 70 =4,900 subtracted from 6,328=1,428. Has it. 1,428 into 3 squares equals 32 +20 +2 , 6,328=70 +32 +20 +2 ."
This took him 10 seconds.
Doubtless all calculators who are conscious of how they operate use shortcuts familiar to the rest of us (such as multiplying by round numbers close to those in the problem and then correcting) but perhaps more boldly -- and quickly. Aitken was asked once by his children to multiply 123456789 by 987654321.
"I saw in a flash that 987654321 by 81 equals 80,000,000,001, and so I multiplied 123456789 by this, a simple matter, and divided the answer by 81."
The answer was 121932631112635269. He had it in about 30 seconds. The part that gets you is that "flash" in the first sentence. It is not a flash that comes to most of us, even if we know the formula Aitken used, ab=(ac)b/c.
Aitken, a New Zealander who taught math at the University of Edinburgh and died in 1967 at 72, was not great at numbers as a boy until a teacher showed him the uses of the formula a -b =(a+b)(a-b) in squaring a number.
"Suppose you had 47 -- that was his example -- he said you could take b as 3," Aitken told an investigator. "So (a+b) is 50 and (a-b) is 44, which you can multiply together to give 2200. Then the square of b is 9 and so boys, he said, 47 squared is 2209. Well, from that moment, that was the light, and I never went back. I went straight home and practiced and found that this reacted on every other branch of mathematics. I found such a freedom."
There are all sorts of tricks. Cube roots are a favorite with performing calculators because, Smith writes, they're easier, "since the last digit of the power unambiguously determines the last digit of the root." Fifth roots, he insists, are also duck soup.
"The difficulty of extracting the root of a perfect power has little to do with the size of the power involved. Much more important are the number of digits in the root and the particular power selected." So he says.
Following the techniques by which hundred-digit numbers and other unwieldy propositions can be broken down and manipulated -- from factoring to logarithms to perpetual-calendar formulas -- the lightning-minded turn Brobdingnagian problems into merely enormous ones.
But what about those others, the ones like Devi, who says the answers just pop up in her head (even, sometimes, when she is starting to protest that she doesn't feel like working)? What about the retarded ones, such as the much-discussed inmate of Marcy State Hospital in Utica, N.Y., who could instantly give days of the week for any date but not much else? Or the African slave Thomas Fuller, who could multiply nine-digit numbers but never went to school?
Smith has a fascinating answer: Maybe this extraordinary ability is like language. All of us, even the near-illiterate, have unconsciously absorbed an amazingly complex set of rules in speaking our native tongues. The subtleties of grammar take years to master in a second language, yet children weave their way through these nuances without a thought.
An example from Smith's book (published by Columbia University Press in 1983, by the way, and a compulsive read even for a math klutz):
You can say, "I wonder why Fred is in Europe." Or you can put it, "I wonder why Fred's in Europe."
You can also say, "I wonder where Fred is in Europe."
But you can't say, "I wonder where Fred's in Europe."
You know that. I know that. We all know that. You just can't say it. But (except for a few scholars) we couldn't tell you why.
And nobody taught us, either, in so many words. What's more, we know it instantly. It doesn't even come in a flash. We simply know it.
Why is that? What method are we using?
Smith suggests that our unconscious facility with language is related to the math prodigy's facility with numbers, for numbers are like a language which is the prodigy's native tongue but is for the rest of us a second language.
Similarly, more than memory is involved. "Some process," he writes, "that goes beyond mere memorization is clearly at work in language ... Some sort of processing, akin to calculation, is involved."
That's still not enough to explain all the strange and beautiful things that go on in the minds of mathematical prodigies.
These are special people, set apart in ways we don't always notice. Their lives are different, too, in the first place because of their unusual talents. Shakuntala Devi was born in Bangalore, India, to a Brahmin who had run off with a circus to walk the tightrope and later to be shot from cannons. The prodigy was born when her father was 61 and her mother 15. At 3, Devi began touring with her father's magic show, a spinoff from the circus.
When he was sick she would do his card tricks, but instead of prearranging and false-shuffling, she would simply memorize all 52 cards in order. At age 5 she watched her uncle working out a cube root problem for his amusement. She told him the answer as soon as she saw the problem. He shushed her, of course, and went on figuring. Ten minutes later he came up with the same number ... and turned to her with wild surmise.
Buxton spent all his life in a Derbyshire village, an illiterate farm laborer who did staggering sums for free beers at the local pub -- the number of barleycorns it would take to make a row eight miles long, for instance. He had his own names for large numbers: a "tribe" was 10 to the 18th, a "cramp" was 10 to the 39th, and so on.
Zerah Colburn of Vermont was exhibited all over Europe and America, tested by such scientists as Faraday and LaPlace, painted by Rembrandt Peale and Samuel Morse. He had 12 fingers and 12 toes.
One day when he was 5 his father overheard him reciting the multiplication tables, though he had only just started school and couldn't read or write numbers. Soon he was squaring four-digit numbers in his head and solving complex arithmetic problems. At 6 he could, say, list all six pairs of factors for 1242 (54-23, 9-138, 27-46, 3-414, 6-207, 2-621) as fast as he could say them.
Treated as a sideshow freak, he didn't begin his education until he was 11, and then only at the insistence of Washington Irving. An actor, author and minister, he was excoriated by apparently envious critics for conceit before he died at 35.
Some prodigies were widely celebrated and respected, some were used like trick dogs, some lived and died in obscurity, local characters who could calculate in their head whatever fantastic problem you could devise -- the area required by 3584 broccoli plants -- or rattle back at you as long a list of numbers as you could give. And some were so mentally dim that they couldn't take care of themselves.
A few, notably the artist Nadia and the famous twins, John and Michael, were "cured" of their gift by therapy in the name of convention and social acceptability. Nadia, no longer autistic, talks but has lost interest in sketching. The twins, who have graduated from their mental institution to a halfway house, have been separated and now can do menial work but, as neurologist-author Oliver Sacks observed in his touching piece for The New York Review of Books nearly three years ago, "they seem to have lost their strange numerical power, and with this the chief joy and sense of their lives."
The twins were 26 when he visited them, had been diagnosed as severely retarded or autistic and dismissed as not very remarkable idiot savants.
They remembered numbers of 300 digits and more, did the day-of-the-week trick and also could recount the events of any day in their lives from about age 3 on.
"A box of matches on their table fell and discharged its contents on the floor: '111,' they both cried simultaneously; and then, in a murmur, John said, '37.' Michael repeated this, John said it a third time and stopped. I counted the matches -- it took some time -- and there were 111."
And 37 goes into 111 three times.
They just saw it, they told Sacks.
Later, he found them "seated in a corner together with a mysterious secret smile on their faces," saying six-figure numbers to each other. Sacks finally figured out that the numbers were primes. He got hold of a book of prime numbers and joined them at their game, reciting an eight-digit prime.
"There was a long pause -- the longest I had ever known them to make, it must have lasted a half-minute or more -- and then suddenly, simultaneously, they both broke into smiles."
Then they topped him with a nine-digit prime, and he topped them with a 10-digit one, to their renewed wonderment. But that was as far as Sacks' book went. Soon Michael and John were trading 20-figure primes while he gaped at them.
"There is no simple method of calculating primes," the author adds. "There is no simple method for primes of this order -- and yet the twins were doing it nonetheless."
It is a commonplace that music and math are profoundly intertwined, and Sacks speculates that the twins experienced some sort of harmonic pleasure in the very thought of primes. (One is reminded of Fibonacci numbers, a sequence that mathematically expresses the peculiar "rightness" of proportion in the chambered nautilus and innumerable other natural forms.)
"They do not approach numbers lightly, as most calculators do," Sacks writes of these brothers. "They are not interested in, have no capacity for, cannot comprehend, calculations. They are, rather, serene contemplators of number -- and approach numbers with a sense of reverence and awe. Numbers for them are holy ..."
Study, analyze, theorize, explain, this we do, and more. The mystery remains.