Русская версия Mnemonic - Articles
Techniques of Juniors N7 1974 г.


Mental calculation
Easy arithmetical puzzles our life ask us every minutes: more or less all of us can calculate mentaly. Methods of calculations "on mind" uncomplicated and described at the last century by Sergey Aleksandrovich Rachinsky, great scientist and remarkable teacher, author first in Russia tasks-book on "mental count". Apropos, S. A. Rachinsky painted like school teacher by his student, known paintist Nikolaev Bodanov-Belsky in the picture "Mental calculation".
Amazingly, but noncunning exercise of schoolboys of younger classes can become a subject of the rapt attention of scientists. Phenomen of superquick calculating, possibility to handle in the mind multi-digit numbers at the speed of PC - here is that forces specialists in the brain to find and research the people, possess giving abilities to the mental calculation.

PRODIGY - CALCULATORS

Viktor Pekelis


Two thousand besoms, five hundred golik, on three coin a hundred - how much roubles?
Old russian riddle

No one possibilities of our brain does seem such amazing, as a riddle a prodigy-calculators.
...On the music hall redeem light. On the stage, brightly refresh by lights of footlights, come a person in the strict black suit - not circus actor, not master of ceremonies, not performer popular songs. Beside a chalk and rag in hands. He looks unaccustomed on the stage.
Show begin. Hundreds of spectators with unflagging attention keep a check on performer.
- Call me, please, - ask artist to spectators. - two multi-digit numbers and, please, find with me together their product.
- One million five hundred ninety four thousand three hundred twenty three multiply by three thousand four hundred fifty six, - ask from the common-room.
Passes several seconds, and all read on the board a result - 5 509 980 288.
Actor patiently waits, while spectators multiply numbers on the paper. After this he call also all results, received during the multiplying.

Wim Klein, human-computer of CERN.
What this talent is? No description, no tale are to give us full presentation. Needed to be present under alive demonstrations, to understand how equitable epithet "prodigy".
Here is tale on the experiment, maded one of the researchers with Mlle. Osaka. She was asked raise in the square 97, get a tenth power of same number. She do this immediately. Then offerred to extract a sixth root from 40 242 074 782 776 576. She answerred immediately and without mistakes.
At 1927 a doctor Osty and mathematician Sent-Lage examined a blind calculator Lou Fleury. Amongst problems was the following: is given number, needed to distribute its on the cube of certain number and four digit number.
Fleury get number 707 358 209. He thinking 28 seconds and give answer: 891 on cub and 5236. He get 211 717 440. Answer was after 25 seconds: 596 on cub and 8704.

Aron Chikvasvily
On Vansky region of Western Georgia live Aron Chikvashvily. He easy manipulated on mind with big numbers. "Counting mechanism" of Chikvasvily do not know taed and errors.
One time his friends want to check possibilities of prodigy-calculator. Task was difficult: how many words and letters will say commentator, commented second round football match "Spartac" (Moscow) - "Dinamo" (Tbilisy). Simultaneously was put tape-recorder. Answer comes as dictor say last word: 17 427 letters, 1835 words. On checking goes... five hours. Answer was correct.
39-years Aron Chikvashvily graduated jurist and economic faculty of Institute.

Phenomenal gift to counting was finding from Lidoreau on three years old, when he can not read and write.
Amongst the prodigy-calculators big popularity use problems, in the base which lies a calendar calculus. Mentally flying through age and millenniums, overcoming difficulties of nondecimal correlations (week consists of 7 days, day from 24 hours, hour from 60 minutes etc.), they, are in a matter of seconds capable to do to hundreds the operations and report that 1 January 180 year was a friday. And all this is done with provision for leap years, changing a calendar at 1582 years etc. They, for instance, can say how much seconds pass from a time of deaths Neron till falling Konstantinopol. One time on the conversation two calculators Inaudi and Dagbert jokingly asked each other by questions of such sort: what weekday will 13 Octobers 28448723 year?
Some problems, which people-counters easy solve on couple of seconds, in the opinion of mathematicians, have require many months usual calculating. After this need a long time to check received results or have recourse to electronic machine.
What methods use a prodigy-counters? Comes "gift" from childhood, in adolescence or is gain, developed during life?
Torture to explain this ability by the exclusive memory, that psychologists name "hypermnesya". Certainly, we have here a manifestation of monstrous memory, but one memory can not to explain this phenomen.
Tell that Gauss father usually paid to workers at the end of week, adding to the daily earnings a charge work after main time. One time, after Gauss-father has finish calculations, a child who keep a check on operations father, which was only three year, sad:
- Father, answer not correct! Here is real sum.
Calculations have repeat and with the wonderment are convince that tot has indicate a correct sum.
Several years ago newspapers reported on the young mathematical phenomenon Borislav Gadzhansky.
- Can you, Borislav, extract a root twenty second power from the number 348 517 368 454 361 458 872?
After minute of thinking boy sad: "Eight".
- And now calculate a root thirty first power from the number 538 436 517 832 435 456 582.
One more minute for thinking.
- Four.
In eleven years Borislav Gadzhansky from the Jugoslavian city Zrenyane fine knew a high mathematician in the volume of program Institute and unassisted pencil and paper produced the most complex mathematical calculations.

At begining of this centyru on Russia was very popular "mathematician on music-hall" R. Arrago.
Show this gift much early or much late, its appearance always spontaneously. Occurs momentally conversion. Owner of gift sometimes cans be "retarded" in all other objects, but between numbers he feels itself as at home and quickly reaches fantastic virtuosity.
What happend with prodigy-counter on future?
Usually their skill is infinitely improve up to deep old age. But can be, that time to time it disappears, on the measure that as its owner gets usual for all children a education. For instance, Ampere become one of the most famous scientists but lost his ability to mental count, until develope cnoledge on classical mathematic. On the contrary, Gauss and Euler connected up to deaths both sides they geniouslity.
Interesting, that many people-counters had not understandings how they do it: "Calculate, and all! But how calculete, god knows". Such answers are not amazing. Some counters were quite uneducated, people. Englishman Buxton, counter-virtuos, so never and did not learn to read, did not know numbers. American negro a counter Thomas Fuler died ungrammatical at an age 80.
Such people always much interested psychologists and mathematicians, which tried to realize, in than secret of their abilities. But explanations, which prodigy-counters gave, try to reveal its skill, at first thought seemed strange, and even much.
For instance, Urania Diamandi told - have numbers she helps their colour: 0 - white, 1 - black, 2 - yellow, 3 - scarlet, 4 - brown, 5 - blue, 6 - dark-yellow, 7 -purpul, 8 - ser-blue, 9 - dark-бурый. Process of calculation was presented by her in the manner of endless symphonies of colour.
Mondeu and Colburn clear saw, as before their eyes are form up numbers of numbers, written whose-that invisible hand. Their "acceptance" was conclude in that, to read this "magic" record. Brother Urany, Periceles Diamandy, told: "Numbers as it were agglomerate beside me in my brain".
Much "easy" Inaudi method. He seemed, is wake instead of he considers whose-that voice, and, while this internal voice produces calculations, itself it or continues to talk, or play music. Mourice Dagbert produces dizzy calculations, playing on the violin.
Several years ago in France, in Lill, in whitness of the authoritative jury from physicists, engineers, cyberneticses, mathematicians and psychologists Maurice Dagbert has enter in the dispute with the electronic calculating machine, produce near million operations at a second.
Dagbert has declare that will acknowledge itself win in that event only, if machine solve seven problems earlier, than he solve ten...
Dagbert solved all 10 problems for 3 minutes 43 seconds, but electronic machine only for 5 minutes 18 seconds!

Igor Shelushkov.
Such competitions not easy. I most recently conducted them in the Institute of cybernetics of Ukrainian academy of sciences. In the competition participated a young counter-phenomenon Igor Shelushkov, graduate student of Gorky Pollytechnic Institute (now he already teacher of this institute and is prepare to protect a thesis) and electronic computing machine "Mir".
I should say several of words about machine. It can solve a many equation system of single-line programming problem, calculate network graphs - in general, execute a number of complex mathematical operations. Machine its creators call "evaluator with the higher education". Not only that it remembers 12 thousand symbols (7 pages of text) and quickly counting. In it "from giving birth" are mortgaged main formulas, which us taught in the school and Institute.
As you see partner serious.
Judged a duel a people authoritative: mathematical programming department head - a professor and its employees.
I do not know, as on competitions in France, but here created equal conditions for the person and machine. This is because many problem machine calculate quicker than person. But there such, that person in general not under power. In the Institute cyberneticses have select corresponding problems, have define moments their "entering" for the person and for the machine, necessary accuracy of deciding - before what sign and etc.
Necessary pay talent of Shelushkov. He brilliantly won competition, either as Dagbert in France.
Last time a prodigy-counters still compete with machines, but all less use abilities for demonstrations their public. Their more interest a practical using a talent and scientific work. Dagbert, for instance, concerns with a mathematics, but Shelushkov will teach, prepares a thesis.

Shakuntala Devi
In Sydney's university in India too passed competitions of person and machine. Shakuntala Devi too has overtake several computing machines. She too wants to bring a practical profit. It helped Indian banks to reconcile and reduce billionth balances, conduct enourmous calculations, which will help for solving complex demographic problem for India.
Some prodigy-counters were subjected to a scientific examination. Inaudi was once invite on the meeting of French academies of sciences. Report on the meeting was given by the mathematician Darby. Scientists come to a conclusion, that Inaudi uses some classical acceptance, which it itself "has re-open". One of the commissions at academies, in which, in particular, enterred known scientific Arago, Koshi, researched Anry Mondeux. On the certificate Koshi, half-literate son дровосека Mondeux used a binomial theorum. To similar conclusions come an academy and at the experiment at 1948 years with Mourise Dagbert.

By reading article about mathematician-actor Арраго, labour of Melekess plant Josef Prikhodko suddenly understand he can perform such show. "First, who introduse about my abilities, - tells Josef, - was my wife. She do not believe me". Now Prikhodko famoust mental-calculator.
Scientists consider that gift to phenomenal calculatings as it observe beside adult counters, is in some degrees gift "trained" (obtained as a result of systematic exercises). Strolling on number jungles, people-counters usially find acceptance, which give it a possibility to shorten calculations.
Probably, single scientifically motivated and sufficiently in detail developping system of sharp raising a mental countings was created during Second World War by Zurich professor mathematicians I. Trachtenberg. It is known under the name "System of quick calculating".
History of its making unusual. At 1941 an fascists threw Trachtenberg in prison. To bacame live in nonhuman conditions and save normal his phyche, Trachtenberg begin to develop principles of mental calculations.
For four terrible years of stay in prison professor to manage to create a graceful system of speed educating a children and adult bases of mental calculations.
After the war Trachtenberg created and led Zurich mathematical institute, get world popularity. System of Trachtenberg allows sharply to accelerate a process of performing the operations of multiplying, dividing, adding, squaring and rooting.
As we see, fast count not a secret for seven seal, but scientifically developped system. If there are a system, than its possible study, follows, and use.

For all who interesting this problem, mentioned on article, we recomended read next books:
1. Мелентьев П. В., Быстрые и устные вычисления. М., "Гостехиздат", 1930.
2. Гольдштейн Д. Н., Техника быстрых вычислений. М., "Учпедгиз", 1948.
3. Катлер Э. и Мак-Шейн Р., Система быстрого счета по Трахтенбергу. М., "Просвещение", 1967.
4. Пекелис В. Д., Твои возможности, человек! М., "Знание", 1973.
5. Фаермарк Д. С., Задача пришла с картины. М., "Наука".