The Great Mental Calculators: The Psychology, Methods, and Lives of Calculating Prodigies, Past and Present
In Brunswick, Germany, in 1780, a stonemason was calculating the wages due his workmen at the end of the week. Watching was his three-year-old son. “Father,” said the child, “the reckoning is wrong.” The boy gave a different total which, to everyone’s surprise, was correct. No one had taught the lad any arithmetic. The father had hoped his son would become a bricklayer, but the boy, Carl Friedrich Gauss, thanks to his mother’s encouragement, became one of the greatest mathematicians in history.
Regardless of such anecdotes, the ability to calculate swiftly and accurately in one’s head seems to have little correlation with creative mathematical ability or high intelligence. Some eminent mathematicians—Gauss, John Wallis, Leonhard Euler, John von Neumann, to name four—had this ability, but most first-rate mathematicians were and are no more skilled in mental arithmetic than good accountants. A few calculating prodigies have even been mentally retarded. No one knows the extent to which this curious skill is genetic, or how much is the result of environment and arduous self-training.
Steven B. Smith, in his admirable history of calculating prodigies—the best, most comprehensive, most reliable book yet written on the subject—thinks that the talent springs mainly from strong childhood motivations. For a variety of reasons, not well understood, a child, often isolated and lonely, will fall passionately in love with numbers. “Children need friends,” Smith writes, “for amusement and companionship. They often devise imaginary friends to keep them company when flesh and blood friends are absent. Calculating prodigies have made numbers their friends.”
When you and I see a license plate on a car ahead of us, we usually see a meaningless number, but to calculating geniuses it is invariably rich in properties and associations. If it happens to be prime (a number with no factors except itself and I) they will instantly recognize it as a prime. If composite (nonprime), they may at once determine its factors. Consider 3,844. “For you it’s just a three and an eight and a four and a four,” said William Klein to Smith, who considers Klein the world’s greatest living mental calculator. “But I say, ‘Hi, 62 squared.’ ”
There is an interesting parallel, Smith suggests, between mental calculating and juggling. Almost anyone can learn to juggle, but only a few are driven to practice until they become experts, and even fewer make it their profession. Children who learn to juggle numbers in their head diverge in later life along similar paths. Some lose interest in the art, some preserve it as a hobby, some make good use of it in their careers. On rare occasions, when talent and passion are high and environmental influences appropriate, a young man or woman will work up a “lightning calculation” act and go into show business.
The stage calculators—like magicians, jugglers, acrobats, tap dancers, chess grandmasters, and pool hustlers—are a diversified breed, with almost nothing in common except their extraordinary ability. Consider Zerah Colburn, one of the earliest and fastest of the calculating wizards. Born in 1804, the son of a poor Vermont farmer, he was only six when his father began to exhibit him. Within a few years he became a celebrity both here and abroad. Washington Irving helped raise money to send Zerah to school in Paris and London. After his education, he gave up his stage career to become a Methodist preacher.
Before Colburn died, aged thirty-five, he wrote a quaint autobiography with the title: A Memoir of Zerah Colburn; written by himself—containing an account of the first discovery of his remarkable powers; his travels in America and residence in Europe; a history of the various plans devised for his patronage; his return to this country, and the causes which led him to his present profession; with his peculiar methods of calculation. Aside from his calculating prowess, the only other notable aspect of the man was that he was born with six fingers on each hand and six toes on each foot, like the giant of Gath (with whom he felt a kinship) mentioned in the Old Testament (I Chron. 20:6).
The chapter on Colburn is one of the most fascinating in Smith’s book. The rustic youth’s methods were discovered by himself, but unlike many stage calculators he did not mind explaining them. When the Duke of Gloucester asked how he had so quickly obtained the product of 21,734 and 543, Colburn said he knew at once that 543 was three times 181. Because it was much easier to multiply by 181 than by 543, he first multiplied 21,734 by three, then multiplied the result by 181.
While Colburn was becoming famous in America, his counterpart in England, George Parker Bidder, was on tour as a nine-year-old calculating prodigy. The two boy wonders eventually met for a contest in Derbyshire (Colburn was fourteen, Bidder twelve), but there was no clear victor. After an education at the University of Edinburgh, Bidder became a first-rate civil engineer, retaining his calculating powers throughout a long, happy, and productive life.
Shortly before he died, Smith tells us, Bidder was visited by a minister who had a strong interest in mathematics. From the nature of light, Bidder told his guest, one could gain insight into both the largeness and the smallness of the universe’s structure. Light travels, he said, at 190,000 miles per second (the best estimate of the day), yet space is so vast that it takes light an enormous time to go from star to star. At the other end of the scale, the wave length of red light is so small that 36,918 waves extend only an inch. The minister wondered how many waves of red light would strike the eye’s retina in one second. “You need not work it,” said Bidder. “The number of vibrations will be 444,433,651,200,000.”
Among this century’s professional mathematicians, the greatest all-around mental calculator was Alexander Craig Aitken, professor at the University of Edinburgh and the author of several textbooks and some eighty papers. His lecture in 1954, “The Art of Mental Calculation,” is the richest source in print on the psychological processes involved in the art.
As Aitken said in his speech, and Smith stresses in his book, calculating prodigies fall roughly into two groups: those who “see” numbers in their minds, and those who “hear” them. Auditory calculators, such as the Dutchman Klein who can multiply any two ten-digit numbers in about two minutes, usually accompany their mental labors with muttering, or at least lip movements. Visual calculators stare silently at the written numerals or off into space. Aitken couldn’t decide whether he was visual or auditory. Here is how he put it:
Mostly it is as if they [the numbers] were hidden under some medium, though being moved about with decisive exactness in regard to order and ranging; I am aware in particular that redundant zeros, at the beginning or at the end of numbers, never occur intermediately. But I think that it is neither seeing nor hearing; it is a compound faculty of which I have nowhere seen an adequate description; though for that matter neither musical memorization nor musical composition in the mental sense have been adequately described either. I have noticed also at times that the mind has anticipated the will; I have had an answer before I even wished to do the calculation; I have checked it, and am always surprised that it is correct.
Like all lightning calculators, Aitken had a prodigious ability to memorize long strings of digits. There are ways to do this by clever mnemonic systems—translating groups of digits into picturesque words, then joining the words by outlandish images—but such techniques are much too slow for rapid calculators. In doing mental multiplication, for example, partial products have to be fixed in the mind until the process is completed in just a few seconds. Aitken mentioned in his lecture that he once amused himself by memorizing pi to a thousand decimal places. Smith tells how Aitken interrupted his talk to recite this long chain of patternless digits, while someone checked its accuracy—there were no errors—against a table of pi. Hans Eberstark, an Austrian mental calculator, has memorized pi to more than 10,000 places, and Smith cites others who have gone far beyond that.
Aitken distrusted all mnemonic tricks. “They merely perturb with alien and irrelevant association a faculty that should be pure and limpid.” His way of memorizing pi was to arrange the digits in rows of fifty each, then divide each fifty into ten groups of five, and “read these off in a particular rhythm. It would have been a reprehensibly useless feat, had it not been so easy.” Interest in the sequence makes the task of memorizing much easier, Aitken added. “A random sequence of numbers, of no arithmetical or mathematical significance, would repel me. Were it necessary to memorize them, one might do so, but against the grain.”
In the demonstrations of calculators like Aitken who are not in show business, there is no need to deceive, but when we turn to the acts of the vaude-villians, where the purpose is to astound and entertain, there are overwhelming temptations to use subterfuges that make the show even more impressive. Smith covers them all. Suppose, for instance, a stage performer has asked his audience for two five-digit numbers. He may say, “Will you please repeat that last number? I’m not sure I heard it correctly.” While saying this, and while the number is being repeated, he has already started multiplying in his head.
The performer may slowly chalk the two numbers on a blackboard. By the time this is finished, additional seconds for calculation have been gained. A reporter seeing him write the product immediately after writing the two numbers will understandably be convinced that the performer obtained the sum in two seconds, not realizing that the actual calculating time was closer to two minutes. Moreover, the best methods of multiplying large numbers in the mind build up the product from left to right. This gives the performer still more time. While he is writing the product left to right, he is continuing to calculate digits near the end.
Another important secret of lightning calculation, though stage performers often falsely deny it, is that early in life they have memorized the multiplication table through 100. Thus in operating with large numbers, they can handle the digits by adjacent pairs and cut calculating time in half. Moreover, thousands of large numbers of the sort that come up often in audience questions, such as the number of seconds in a year, or inches in a mile, or the repeating sequence of digits in the decimal form of 1/97 (it has a period of 96 decimals), can be committed to memory. No stage performer would ever say, “Please don’t ask me that, because I already know the answer.”
Some stage calculators are not beneath planting confederates in the audience to call out problems for which the answer is already known. Smith thinks this occurs rarely, but I am not so sure. I know many magicians who do what the trade calls a “mental act”—feats that purport to be accomplished by psychic powers. Their use of secret accomplices is quite common. Why should show-biz calculators be less deceptive?
In some lightning calculation tricks, the use of an accomplice is cleverly hidden by the fact that only part of a problem need come from a confederate. A marvelous example of this is provided by Smith’s account of a performance in 1904 at the University of Indiana by a calculator who called himself “Marvelous Griffith.” Griffith wrote 142,857,143 on the blackboard, a number probably called out by a confederate. A second nine-digit number was then supplied by a professor whom everyone knew could not be an accomplice. While the professor was still writing his number, Griffith began chalking the product of the two numbers from left to right. When it was found to be correct, the audience stood up and cheered.
As Smith points out, although Griffith was a skilled mental calculator, this feat can be done by anybody. You have only to divide the second (legitimate) number by seven, and do this twice. If there is a remainder after the first division, carry it back to the initial digit and divide through again. If the final division doesn’t come out even, you goofed. The trick works because 142,857,143, is the quotient when 1,000,000,001 is divided by seven.
Unfortunately, the numbers involved in this little-known swindle are beyond the capacity of pocket calculators, but if you want to astound your friends there are simpler tricks based on the same principle. For example, the product of 1,667 and any three-digit number abc can be obtained mentally as follows. Add zero to the end of abc, then divide it by six. If there is no remainder, continue by dividing abc by three. If the remainder after the first division is two or four (it cannot be anything else), carry half the remainder (one or two) back to the first digit and divide abc by three.*
Most stage performers include in their act two demonstrations that are much easier than they seem: giving the roots of perfect powers, and giving the day of the week for any date called out. Only small charts need be memorized for finding cube roots (they are easier to calculate than square roots), and fifth roots are still simpler because their last digit is always the same as the last digit of their fifth power. You’ll find the details in Smith’s book, as well as an excellent system (there are many) for doing the calendar trick. Both feats can be easily mastered by anyone who cares to spend a little time practicing.
Jugglers, by the way, also are not beyond augmenting their skill with fakery. It is possible, for instance, though not easy, to spin two large rubber balls, one on top of the other, on a finger. It is absolutely impossible to spin a stack of three. Nevertheless, the great Italian juggler who worked for the Ringling Brothers Circus under the stage name of Massimiliano spun three balls in his act. His secret was to use a ball in the middle that was weighted on the bottom; in effect, he was spinning only two balls.
Because showmanship is the essence of good vaudeville, one can forgive professional jugglers and acrobats and calculators for highlighting their acts with harmless flimflam, but it means that in reading accounts of their marvels one must often take them with a grain of salt. You don’t have to believe that Unus, a circus acrobat, can actually do a handstand on one gloved finger even if you see him appear to do it. You don’t have to believe, when you read in the 1984 Guinness Book of World Records, that Shakuntala Devi of India (one of the few women calculators currently performing) multiplied two thirteen-digit numbers, each randomly selected by a computer, in twenty-eight seconds. As Smith politely understates it: “Such a time is so far superior to anything previously reported that it can only be described as unbelievable.”
Do lightning calculation acts have a future, or has the computer rendered them uninteresting? At the close of his lecture, Aitken admitted that his own abilities began to decline when he got his first desk computer. “Mental calculators, then, may, like the Tasmanian…be doomed to extinction,” he said. “You may be able to feel an almost anthropological interest in surveying a curious specimen, and some of my auditors here may be able to say in the year 2000, ‘Yes, I knew one such.’ ”
Perhaps Aitken was right. On the other hand, the electronic calculator has in certain respects increased the entertainment value of such shows. One of the things that slowed up past performances was the inordinate amount of time required to verify large calculations. Frequently persons in the audience would offer a problem they had previously calculated incorrectly, and much time would be lost in setting the record straight. Today, mental calculations with big numbers can be verified quickly by anyone with a small computer. As young Arthur Benjamin has discovered—he is the only American now performing a mental calculation act—this makes possible many entertaining feats that were not available to earlier performers.
The use of computers also raises a truly dreadful possibility. So far as I know, it has not yet been exploited on the stage, though I would expect it to be eventually. There are now simple devices for wireless communication between two persons, with receivers tiny enough to be concealed in the ear or the anus to provide easily heard or felt beeps. There is nothing to prevent a confederate backstage, or even sitting in the audience, from quickly solving a problem on a computer, then secretly relaying the answer to a performer, who may even be a horse or dog, by operating a switch with his toes inside a shoe. If this ever becomes a common practice, the honorable art of rapid mental calculation will have indeed deteriorated to the level of a calculating animal act, or the acts of mediocre magicians who pose as psychics with awesome paranormal powers.
For other magic numbers of this sort, and why they work, see the chapter on lightning calculation tricks in my Mathematical Carnival (Knopf, 1975).↩
For other magic numbers of this sort, and why they work, see the chapter on lightning calculation tricks in my Mathematical Carnival (Knopf, 1975).↩