The above title is also that of a new book by Steven Smith. Naturally, the book is full of anecdotes about the phenomenal accomplishments of calculating prodigies, both unlettered children and such famous scientists and mathematicians as Euler, Gauss, and A.C. Aitken. The latter

"...had the uncanny power of mentally computing, to a long string of decimals, the values of e and e^{163} When asked (by his children) to multiply 987...1 by 123...9, he remarked afterwards: 'I saw in a flash that 987...1 multiplied by 81 equals 80 000 000 001, and so I multiplied 123...9 by this, a simple matter, and divided the answer by 81.'"

But what, asks Smith, led Aitken to 81? To this question, which is the heart of the mystery, he commendably admits he has no reply. And the same deep mystery confronts us even after all has been said about the sur, as distinct from the underlying, structure of the processing.

At the unlettered end of the spectrum of mental calculators, the

"...ignorant vagabond, Henri Mondeux, who at the age of 14 years, before the French Academy of Sciences, was able promptly to state two squares differing by 133."

Of course, some mental feats of calculation can be done consciously employing various shortcuts and mathematical tricks. The really fantastic performances, however, are accomplished unconsciously. No one knows how, even the calculators themselves.

(Cohen, John; "What Makes a Calculating Prodigy?" New Scientist, 100:819, 1983.)