Trade Secrets

To the Editors:

I was very pleased with Martin Gardner’s generous and perceptive review of my book, The Great Mental Calculators, but I hope that his extended discussion of deception in mental calculation will not lead readers to conclude that all the feats of professional calculators are trivial or fraudulent.

As Gardner remarks, some calculations are less difficult than they appear, such as extracting the roots of odd perfect powers (e.g., what is the twenty-third root of some 41-digit number where the root is known to be an integer). Such problems must be simpler than they seem, because they seem impossible.

In fact, these problems are trivial when the answer contains no more than two digits (as in the case above), but when the answer contains, say, eight digits, the difficulties are enormous. When Wim Klein extracts the thirteenth root of a hundred-digit number he must determine the logarithm of the first four digits of the power, divide it by thirteen and obtain the antilog. This gives him the first five digits of the root. (Klein has memorized the logs to five digits of the first 150 integers. The rest he gets by factoring, adding together the logs of the factors, and by extrapolation.) Then, to fix the last three digits of the root (in the case of an even number), he must divide the entire 100-digit number by 13, retaining only the remainder. His best time for this mental calculation is under two minutes. If you want to get a small idea of the difficulty involved, try dividing a hundred-digit number by 13 on paper as fast as you can and see how often you come up with the same answer.

Of course outright cheating occasionally occurs, but it is usually apparent to the knowledgeable observer. If someone appears to perform an evidently impossible calculation, they had best be able to give a credible account of how it can be done or I will start looking for tiny transmitters, concealed calculators, advanced knowledge of the problem, peculiarities in the numbers, and so forth. I do hope that the chicanery of a few mountebanks will not cast a pall over the legitimate accomplishments of calculating prodigies, and that audiences will become more sophisticated so that they can distinguish between the impossible, the trivial, and the truly phenomenal.

There are also one or two points on which I disagree with Gardner, such as his contention that calculating prodigies in general employ a multiplication table of 100 by 100, but these can be left to the judgment of readers of my book.

Steven B. Smith

Soubès, France

Martin Gardner replies:

I certainly did not mean to suggest that the feats of the great mental calculators are trivial or fraudulent; indeed, I thought my review gave the opposite impression. In any case, I concur with everything in Smith’s letter except his belief that most great calculators did not know the multiplication table to 100.

Consider Wim Klein and A.C. Aitken, two of the fastest mental calculators of recent times. Writing about them in his book Faster Than Thought, B.V. Bowden said: “Both men have most remarkable memories—they know by heart the multiplication table up to 100×100, all squares up to 1,000×1,000, and an enormous number of odd facts, such as that 3,937×127 = 499,999.”

Consider Arthur C. (“Marvelous”) Griffith, a famous stage calculator. William Bryan and Ernest Lindley questioned him at length about his methods, reporting on them in their book On the Psychology of Learning a Life Occupation. Griffith, they write, “has [in his memory] a multiplication table complete to 130—and partial to almost 1,000…is thoroughly acquainted with every prime and composite below 1,500, and can instantly give the factors of the latter.”

Smith acknowledges in his book Klein’s use of a 100 table, but thinks Klein a rare exception. A footnote recognizes Griffith’s claim, but Smith doubts his honesty. On the other hand, Smith does not doubt any statement by a calculator who denied knowing the 100 table.

My contrary opinion rests on the extreme ease with which any mental calculator could memorize such a table, and the enormous aid it would be to him. Fred Barlow, in his book Mental Prodigies, quotes the French mathematician Edouard Lucas: “I formerly knew an instructor whose scholars, of eight to twelve years of age, for the most part knew the multiplication tables extended to 100 by 100 and who calculated rapidly in the head the product of two numbers of four figures, in making the multiplication by periods of two figures.” In my opinion, it would be difficult for a great calculator to avoid memorizing the 100 table.

Smith is impressed by the fact that some calculators of the past, such as George Bidder, denied they knew such a table. I am more impressed by the notorious reluctance of professional calculators, like magicians and locksmiths, to give away all their trade secrets. Of course knowing a 100 table no more implies deception than knowing the 10 table.