Mnemonic - Articles

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Fast calculating Jakov Perelman. 1945

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REAL AND FALSE PHENOMENONS

All who had possibility to be present at show of our mental calculator Arrago, no doubt be amazed his astonishing calculating abilities. Here we see not trick, but rare natural talent. Cube of number 4729, for instance, Arrago calculate near me in the mind less than one minute (result 105756712489), and for multiplying 679321 x 887064, also in mind, he use only 1½ minutes.

I had a possibility to watch a computing work of this phenomenal calculator not only on stage, but in the home situation, from the eye to the eye, and could make sure that no particular computing acceptance he use, but calculate in the mind in general in the same way, as we on the paper. But his unusual memory for numbers helps him work without writing the intermediate results, and fast cleverness allows work with two-digit numbers in the same way easy, as we produce actions with one-digit numbers. Due to this multiplying of six-digit number on six-digit for him problem not much difficulty, than for us - a multiplying three-digit on three-digit.

Such phenomenons, like Arrago here or in the West Inaudi, Diamandi, Ruchle and best of all Dr. Fred Brauns are unic. But alongside with they act special mathematicians of other sort, based they art on one or another arithmetical tricks. You, probably, heared or even be present at shows of "geniously mathematicians", what calculate in the mind with the striking speed, how much you passed the days, minutes, seconds, in what weekday was your birth &c. To execute a most of these calculations, not needed, however, possess unusual mathematical abilities. Necessary only know some secrets of these focuses, - a denouncement which we presently begin.

MEMORIZING NUMBERS

Fast calculator must first of all possess an exceed developed memory on numbers. What super possibilityes reaches such memory beside best calculators, show the following records. Famous german calculator Ruckle learned number, consist of 504 digits, on 35 minutes, but his compatriot Dr. Brown has beat this record, having do same less than 13 minutes!

Of course, such phenomenal memory had from nature only unic ones. Professional calculators, acting on scene, without having such natural memory on digits, help themself by different artificial acceptance (so called "mnemonic"). In usual life we themselves time to time are torture to use similar acceptance, most of what unhappily chosen. With idea, for example, memorize phone number 25-49, we entrust hopes on that number this will easy manage restore in memories, since it is constitute of two exact squares: 25 = 5², 49 = 7². But when come really necessary recall it, we get extensive choice of numbers: 16-25, 36-64, 25-16, 64-16, 81-25 и т.д.

Such failure we get on another situations too. Phone number 17-53 we try to remember, by using fact that first two numbers together (1 + 7) are equal two last (5 + 3). But final not better, than in the previous case. But after all is necessary do not forget, to whose telephone was aplying that and to what other combination. Possible only wonder, as persistently people are torture to use this obviously unimproved acceptance. Passion to it witty has deride a writer J. Gashek in his famous "Adventures of soldier Shweik": "Some time Shweik looks number of gun and finally has say:

- Number 4268. Just same number has one locomotive on Pechka on sixtenth way. Locomotive must be moved to Lissa for repair, but this was not so easy, because the machinist, which must made it, has a very bad memory of numbers. Then chief to distances call him to the chancellery and speaks him: "On 16-м way stands a locomotive N4268. I know you have a bad memory on numbers, but if I will write a number on paper, you paper will lose. But if you stell so weak on the numbers, try to remember that I presently say to you, and you see that possible with lightness to notice any number. Well, so here is. Locomotive, which you is necessary conduct in the railroad yard, is listed for the number 4268. Here is and call attention. The first number - four, second - two. Remember, it become 42, i.e. two on two - four, that gives us first number, and if devide it on two, we will get again two, and thereby beside us we get 4 and 2. Further already simply. Eight, isn't it? Now you move it to memory, that eight in our number is a last. Now you have remember that first numbe - four, second - two, and last - eight. So, we need only to remember a number six before eight. But this is very much easy. Because first numbe 4, second 2, and together will be 6. So number 4268 firmly has seeded beside you in the head. You can also come to this result more easy way, as follows: from 8 subtract 2, will be get 6. Remember: 6. From 6 subtract two, will be get 4. We have already 4 and 68. Now is necessary only between these two numbers to put a number 2, and we get 4268. Possible to do enother, too easy by using multiplyings. Запомните, что дважды 42 равно 84. В году двенадцать месяцев. Надо вычесть 12 из 84, останется 72, и из 72 еще раз вычесть 12 месяцев. Получится 60. Вот у нас уже есть 6, потому что ноль мы можем просто отбросить. How much will two on four? Remember that twice 42 is 84. Per annum twelve months. Is Necessary subtract 12 from 84, will stay 72, and from 72 once again subtract 12 months. Get 60. Here is beside us already be 6, because zero we can simply reject. Значит, если мы напишем 42-6-84 и отбросим последнюю 4, то неминуемо получим число 4268, т. е. номер паровоза, который надо отвести"".

Methods of variety calculators are absolutely other type. Here is one of them, which can be useful for each of us. Calculator involves with numbers determined consonant letters, firmly learn:

numbers	0	1	2	3	4	5	6	7	8	9
letters	Н М	Г Ж	Д Т	К X	Ч Щ	П Б	Ш Л	С 3	В Ф	Р Ц

Так как буквы выбраны только согласные, то их можно, не боясь путаницы, сочетать с гласными, составляя короткие словечки. Например:

numbers	words	numbers	words
1 2 3 4 5	еж яд око щи обои	6 7 8 9	шея усы ива яйцо

Сходным образом составляются слова и для двузначных чисел:

11 - гага 12 - год 13 - жук

14 - гуща 15 - губа 16 - игла и т. п.

Чтобы запомнить число 2549, эстрадный счетчик мысленно подписывает под цифрами соответствующие им буквы:

2 д т

5 п б

4 ч щ

9 р ц

и быстро составляет из них слова, например:

25 дуб

49 ящер

"Дуб" и "ящер" не только легко запомнить, но и связать как-нибудь с фамилией гражданина или названием учреждения, которым принадлежит телефон.

This is one of mnemonic method, used among musik-hall calculators.¹ There are anothers, to which we, however, do not concentrate, and will go over to ways of performing the calculating program numbers.

- I am ... years old. How many days to me? - asked somebody from the public and immediately gets from stage an answer.

- How many seconds to me, if my age ...? - ask another and get answer immediately.

How to made such calculatings?

"HOW MANY DAYS TO ME?"

Problem

Чтобы по числу лет быстро определить число дней, счетчик прибегает к такому приему: половину числа лет множит на 73 и приписывает ноль - результат и будет искомым числом. Эта формула станет понятна, если заметить, что 730 = 365 x 2. Если мне 24 года, то число дней получим, умножив 12 x 73 = = 876 и приписав ноль - 8760. Самое умножение на 73 также производится сокращенным образом, о чем речь впереди.

Поправка в несколько дней, происходящая от високосных лет, обыкновенно в расчет не принимается, хотя ее легко ввести, прибавив к результату четверть числа лет, - в нашем примере 24 : 4 = 6; общий результат, следовательно, 8766².

Прием для вычисления числа минут читатель, после сказанного в следующей статье, не затруднится найти самостоятельно.

"HOW MANY SECONDS TO ME?"

Problem

If age of man who ask is expressed even number not biger than 26, on this question also possible enough quickly to answer, using following acceptance: half of number of years multiply by 63; then same half multiply on 72, result put near by first and prefix three zeroes. If, for example, number of years 24, for the determination of number of seconds you made next: 63 x 12 = 756; 72 x 12 = 864; result 756 864 000.

Like in previous example, here not use leap years - error, what nobody ask calculator, because he show how he work with hundreeds of millions (also it possible improve, by adding number of seconds, corresponding quarter of years).

What based this example on?

Solution

Correctness of our formula is realize much simply. To define a number of seconds, conclude in the given number of years, needed years (in our example 24) multiply on the number of seconds on one year - on 365 x 24 x 60 x 60 = 31536000. We do same, but only big multiplier 31 536 divide on two part (prefix of zeroes is comprehensible). Instead of multiply 24 x 31536, we multiply 24 on 31500 and on 36; but for make it easy calculations change to others, as can seen from the following scheme:

24 x 31536 = {

24 x 24 x

31 500 = 12 x 36 = 12 x

63 000 = 72 =

756 000       864 756 864

We should prefix only three zeroes, and total result is: 756 864 000.

METHODS OF FAST CALCULATING

Мы упоминали раньше, что для выполнения тех отдельных действий умножения, на которые распадается каждый из указанных выше приемов, существуют также удобные способы. Некоторые из них весьма несложны и удобоприменимы; они настолько облегчают вычисления, что не мешает вообще запомнить их, чтобы пользоваться при обычных расчетах. Таков, например, прием перекрестного умножения, весьма удобный при действии с двузначными числами. Способ не нов; он восходит к грекам и индусам и в старину назывался "способом молнии" или "умно-

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¹ Подробнее об этом см. в моей книжке "Фокусы и развлечения".

² Указанными далее приемами ускоренного умножения эти операции облегчаются до чрезвычайности, и миллионный результат получается очень быстро. Советую читателю попробовать произвести то же вычисление и обыкновенным путем, чтобы на деле убедиться, какая экономия времени получается при пользовании указанной формулой и приведенными далее приемами.