Русская версия Mnemonic - Articles

The Great Mental Calculators
By Steven B. Smith. 1983

Chapter 34
Wim Klein
In Marseille on a Sunday afternoon, there was a chap who was eating flames, there was a lady who told the future, there was a chap with a monkey, and I was extracting cube roots. Ho, ho. I'd like to go back and do it again. It was fun. But I did not need to do it. I just got so fascinated with this vagabond life.
Wim Klein, reminiscing about the post war period

Unless otherwise attributed, information in this chapter is based on personal interviews with Wim Klein.
WIM KLEIN (also sometimes referred to as Willem or William Klein) was born in Amsterdam on December 4, 1912 - a Wednesday, as he is quick to tell you.
Klein's interest in calculation began at age 8, when he discovered factoring. "At school we had to factor numbers up to 500. Then I continued on to 10000, 15000, 20000, 25000. As you got so often the same combinations, it is logical that if you know that 2537 is 43 times 59, and you're doing a little show for the godmother of a neighbor celebrating her eighteenth birthday, and they ask you for 43 times 59, you recognize straightaway 2537." Although Klein never set about to learn the multiplication table up to 100 by 100, he gradually acquired it from repeatedly encountering the same combinations. He contrasts this sort of memory, which comes about unbidden simply as a matter of repeated exposure to certain material, to deliberately committing something to memory; the latter he calls "mechanical memory." "The multiplication and the squares up to 1000 I just took as a game. Learning the logarithm table by heart up to 150 is memorizing, but in terms of multiplication I never memorized. It came from the experience I got by factoring - I got often that 2537 is 43 times 59, and 5074 is the double, and 7611 three times the number - so you recognize it."
By exposure he has learned the multiplication tables up to 100 by 100, the squares of integers up to 1000, the cubes of numbers up to 100 and roughly all prime numbers below 10'000. He deliberately committed to memory the decimal logarithms to five places of the first 150 integers. He also knows "some other small things like the first 32 powers of 2, the first 20 powers of three and so on; some logarithms base e, a lot of history; and I also learned by heart the date of birth and death of about 150 composers."
Klein's older brother, Leo, was also an exceptional mental calculator but Wim was the moving force behind the brothers' interest. "Leo was a little infected by me. Because I did it, he also had to do it."
The brothers were, however, altogether different in their methods - Leo's memory was visual, while Wim's is auditory. Leo did not share Wim's fascination for factoring, nor did he care to go beyond three digits by three digits in multiplication.
To illustrate the difference between his methods and Leo's, Klein asked me to call out two three-digit numbers. I chose 426 and 843. He muttered in Dutch and after a few seconds said: "359118" He then took a piece of paper and wrote:
4.26
  X   8.43
12.78
170.40
  3408.00
3591.18
"My brother would say: There's twelve dollars and seventy-eight cents plus a hundred and seventy dollars and forty cents plus three thousand four hundred and eight dollars-three thousand five hundred ninety-one dollars and eighteen cents, which he had to translate into normal pronunciation as three hundred fifty-nine thousand one hundred eighteen.
"But I say hey, 426 divided by 6 is 71, and 843 times 6 is 5058, 5 times 71 is 355 [in this case thousand] and 58 times 71, by experience, without calculating it, is 4,118. You see the difference? If you had taken 427 I should have done quite differently: I should have said '427 divided 7 is 61 and 843 times 7 is 5901; 61 times 5901 is 359961.' You see how it helps using factorization as much as possible? There is a keyhole in a hotel and I've got a key ring with 500 keys. What I have to do is to pick the right key for that slot. For every problem I have to think .straightaway which is the best way-just like a flash."
The difference between the brothers' memory showed up in other areas as well. Leo would come into a town, buy a map, scan it briefly, and know his way about. Klein recalls: "He would say to a waiter in a pub, I have to go there.' The chap would say, 'Well, you go like this.' My brother says, 'And if you go like this, is it not shorter?' The waiter says, yes, sir.' " When the brothers were children and took the streetcar to school, each conductor wore an identification number on his collar. Years later they saw an elderly man selling tickets on the streetcar. Leo asked Wim, "Do you remember him?"
"No."
"In the old days his collar number was 683."
Leo started to talk to the man and after a while he asked: "How many cars have you worked on the tramway?"
"Oh, since.."
"Yes, and years ago you were on streetcar number 60."
"Yes."
"And your collar number was 683."
The old man looked at him: "You damn.. That is correct."
Klein says: "There were hundreds of examples like this. Funny, eh? This visualizing. I cannot do it at all."
Klein's father wanted a successor to his medical practice, so Wim very reluctantly undertook the study of medicine, even though he was already dreaming of show business. In spite of all the time he was forced to devote to studies, he managed to give little shows now and then. He finished his theoretical studies "after a hell of a lot of trouble," but before he could finish medical school his father died, and Wim abruptly halted his studies. "The old man died in 1937. He was always very strict on the penny with my brother and me. After he died, we got a little inheritance. So the brothers took full profit of life. Then the war came. It became really tough, of course, as you may guess. I lost my brother in the war, also killed like hundreds of thousands of others."
For two years Klein worked in a Jewish hospital, as other hospitals were forbidden to Jews. Then the Germans started to take people from the hospitals to camps and Klein was forced into hiding. Leo was not so fortunate; he was sent to a camp in Germany from which he never returned. It is not a period Klein cares to recall. "I had to hide. Some people took care of me. Just say it was like the case of Anne Frank. That is sufficient."
After the war Klein had to find some way to make a living, since the Germans had confiscated all his money. "Everybody needed show business at that moment, after those awful years."
Klein's first post war role as a professional calculator was a non-speaking one. He was decked out as a sort of Indian fakir with turban and a false beard. His partner did all the talking while Klein chalked up the answers.
But the theatrical agency was dissatisfied: "They told me, 'Listen, this act with the beard stinks. Your presentation is vulgar. We will get you in contact with one of the best announcers in the country, and you will travel about with a group of excellent artists.' And then it developed as a really nice act. But then, as Wim Klein was too cheap, they came up with 'Pascal.' In Holland they don't know Wim Klein, they know only Pascal. Why Pascal? It is a French name, but Pascal was the inventor of the calculating machine. When Pascal came to France, where nine out of ten people call themselves Pascal, and my French with my Dutch accent was not good enough, so there I became Wim Klein." (Klein is also sometimes known as "Willie Wortel" - Wortel being the Dutch word for root - from his ability to extract roots of large numbers).
In France and parts of Belgium, Klein had to do his act in his "poor, bad, school French." (He later learned to speak fluent, if somewhat confusing, French, German, and English.) "There was an expression, which in Dutch was a normal expression, so I thought in French it should be the same, so I translated it literally from Dutch into French and it meant something quite different. Nobody told me, but the audience burst out in laughter the whole bloody week. They did not want to tell me what was wrong with it, and only the last day, they told me it meant something like 'How often in the week do you do it?' or something like that. Silly."
During the post war period he appeared in France and Belgium, and began doing radio broadcasts in Holland. While in Brussels in 1949, Klein was down on his luck. "I spent all my bloody money. Then I met some friends and one said: 'I play the guitar and he plays the accordion. You have a blackboard. We make some music and you do some sums.' We went to nightclubs, little pubs. First they made the music, then I came on with my sums, and then the guitarist went around with the hat. After a while we decided to go to Paris. We went to the 'Champs Elysees' where people sit outside, and started to perform, but the police came and said: 'Shut up, you bloody beggars.' My friends went back to Holland, and after that I joined a little cirus. But we went broke. A chap said, 'Why don't you just set up your act at the subway entrance? Lots of people do it.' And yes, it worked perfectly, but there were two enemies - the rain and the police. Some cops would say 'We're here this week, but next week some bastards are coming, so next week go somewhere else.' Or a Dutchman would say, 'Hey, can you fix me up with a nice girl?' 'Yep.' 'Emmanuelle, I've got a guy for you.' 'Here, my dear, this is for you. A tip.' Oh, what great time."
But as Klein had no work permit, the French authorities finally kicked him out of the country. At the train he met a Belgian who had seen him perform several times in the Place Pigalle. The man was living in Mons near the French border. He said that he had contacts in Belgium through which they could organize a lecture tour in the schools. "They put me up a little pension, and they bought me some decent clothes. After three months time I was out of debt."
In 1952 Klein got a job at the Mathematisch Centrum in Amsterdam, where he did various sorts of numerical calculations. "Computers-they didn't exist, or nearly not. I sat in a room with these five heavy-reformist members of the Netherlands Reformed Church, always talking about God and the clergy. I would say 'Good God' and 'God damn it,' and they went to the boss and said, 'Klein is swearing like a docker.' He told them, 'Don't quarrel, let him swear.' They said, 'Yes, but...' So he called me in and said, 'Listen, Klein, quiet down. I know they are idiots, but try to do better.' 'I try, I try very hard, but you know.."
In 1952 Klein began seriously lecturing in schools; again it came about by accident. Whenever important people visited the Mathematisch Centrum, Klein was called upon to give a demonstration, "not as a human computer, but as a human attraction." A French professor from UNESCO saw him there and asked him if he would come to Paris and give a lecture in the Department of Mathematics at the Sorbonne. The planned fifty-minute appearance stretched to two hours. As a result of contacts made, Klein was able to obtain permission to give lectures in grammar schools throughout France. "So I wrote to the Mathematisch Centrum and asked for leave of two months or so. They replied, 'Wim, I'm afraid this means the end - that you will stay longer than two years. But I'm sure you will have tremendous success.'"
In 1954 Klein met the New Zealand mathematician and calculating prodigy, Alexander Craig Aitken, at a mathematical conference in Amsterdam. Later in that year they appeared together on a BBC program. Klein recalls: "He was a lovely man. When Britain had not yet the decimal system, I used to do problems like multiplying £3,7 shillings, 8 pence ha'penny by 29. When I asked Aitken about such problems, he said: 'Oh, I've got enough trouble when I have to fill out my income tax form'
In 1955 Klein toured for nine months as one of the attractions of the "MIRACLES OF THE MUSIC HALL, Starring Some of the Most Unusual People Ever Seen." He was billed as "the man with the £10000 brain. The cast included - the Dare-Devil Denglaros on their Racing Motor Bikes; the Amazing Devero, Escape from a Real Guillotine; Ladd West World's Fantastic Aerial Contortionist; Rondart, Worlds only Dart Blower; the Roller Skating Jeretz from Geneva; Reggie 'yer see' Dennis, Britain's new radio Comedian; the Incomparable Mime Star, Danny O'Dare; and Personal Appearance of The Man Who Was Buried Alive - already seen by 4719329 people in 8 years tour."
Klein reflected on some of his fellow performers: "This chap was a real pig. He always escaped, but we all hoped that he wouldnt. Danny O'Dare the Indian Rubberman, was just a poor devil. With the Moto-devils, every second word was a swear word. You could not speak decently with those people.
"Tommy Jacobson, the armless wonder - the first thing he did was to take a rifle and shoot, then he played the piano with his feet, then the master of ceremonies asked someone to come on the stage for a shave. He told the chap to sit down and said. Tommy, not as much blood as yesterday.' So the poor chap turned his head so, and Tommy took a big knife and ... sccrrr.
"Once Tommy was standing on the platform of a London bus and a chap tried to grab hold of him to pull himself on the moving bus. He grabbed Tommy's raincoat sleeve, but there was nothing in it, so he tumbled into the street clutching a raincoat."
The Miracles of the Music Hall gave two shows a night. After doing his act for the first house, a performer was off until it was time to perform in the second show. Between shows one night, Klein dropped into a pub next door. People from the audience of the first show called him over and started buying him drinks and asking him to do calculations. By the time Klein finally went running back to the theater, the second show was almost over. Someone said to him, "Wim, be careful. The announcer is already in a bad mood." The announcer, who was also the manager, called out, "Wim Klein, there he is, here comes Wim Klein." Klein says, "I was struggling up, you know. He still didn't notice, but then he caught a funny smell of gin. He came really close and said, 'You bloody Peruvian Chinese teapot. If you don't finish your act properly, I'll kick you out straightaway.' I said, 'Yeah, yeah, I've just been kidding.' So I made it, more or less, after the show he said, 'Listen, I also like to drink, but never do it between the two houses. You promise?' 'Yes.' 'Come on, let's have a drink together.' So we both got pissed when the show was over."
After Miracles of the Music Hall, Klein returned to touring schools, in France and then in Switzerland. But by 1957 he decided he wanted to settle down, so he returned to Amsterdam to work at the Mathematisch Centrum. During the summer of 1958 he arranged a two-week tour of Swiss schools. The giant research complex of CERN (European Organization for Nuclear Research) is located near Geneva. Klein was mistakenly under the impression that some of the work of the Mathematisch Centrum was done for CERN, so he decided to telephone CERN while he was in Geneva. He was told to "just pop over." He was introduced to a Dutch physicist, C. J. Bakker. Klein recalled: "We talked and he said, 'Would you like to work in Geneva?' I said, 'That's for other people to say. Is it a possibility for me to get a job here?' And then Professor Bakker said 'Listen, Wim, I cannot decide; I'm only the director general here, you see.' So I said, 'Not so bad.' Naturally, I had a feeling everything would be all right."
An arrangement was made with the Mathematisch Centrum for a three-month leave of absence. After four weeks, CERN asked Klein to stay on permanently. "These three months became eighteen years. That's the CERN story."
In the early days Klein was in considerable demand at CERN. "Computers were not very well developed, and the physicists did not yet program them themselves. From '58 to '65 it was all right for me. And then, it went down, because young physicists did their own programming, and they did not need me as much as before. But the idea to kick me out never came, because of public relations. Very often when physicists came they could not see the machine, someone would say, 'Hey, Wim, do thing for them.'"
Jeremy Bernstein described an encounter with Klein (1963:20): In the summer of 1961, I had an opportunity to work with Mr. William Klein, a programmer and numerical analyst for CERN in Geneva, who must be one of the fastest human computers who has ever lived. I was spending the summer doing physics at CERN and had been working with a friend on a problem. After a week or so, we produced an algebraic formula that seemed admirable to us in many respects, and we wanted to evaluate it. CERN has a large Ferranti Mercury computer, and since at the time neither of us knew anything about programming, we asked for help. Enter Mr. Klein. Mr. Klein is a short, kindly, energetic-looking man in his forties. He is of Dutch origin. He looked at our formula for a few seconds, muttering to himself in Dutch, and then gave us numerical estimates for several of the more complex parts of it. Doing this, he said, helped set up the program for the computer in the most efficient way. I had heard about Mr. Klein's almost incredible ability, and I asked him whether he had considered evaluating our whole expression in his head. He told me that it would involve much too much work and that he was quite glad to turn the job over to the machine. Watching Mr. Klein at work made a deep impression on me.
By the mid-seventies, Klein was growing weary of CERN. In 1975 Amsterdam was celebrating its 700th anniversary, and Klein visited there some eight times. The next year he again visited Amsterdam several times. He decided to retire. "It's too monotonous - 18 years. It was a golden cage, but I prefer silver freedom. So at the end I retired one year before I was 65."
In June 1974, shortly before his departure from CERN, Klein became intrigued with the problem of extracting integer roots of large numbers. The 1974 edition of the Guinness Book of World Records reported that Herbert B. de Grote of Mexico City had extracted the 13th root of a hundred-digit number in 23 minutes.
Klein says: "What is the use of extracting the 13th root of 100 digits? Must be a bloody idiot you say. No. It puts you in the Guinness Book, of course.
I never came on the idea until I got this notice about this man in Mexico. I thought, hey, how interesting. I should have thought of that. First I had to find out how to tackle the problem. Then I needed material - I needed numbers raised to the wanted power. So they wrote a multi-precision program on the computer. And I was practicing like hell, like hell, like hell. Once you know the system for the first one, you have to learn another series of numbers by heart for the next one."
By October 8, 1974, Klein succeeded in extracting the 23rd root of a 200-digit number in 18 minutes, 7 seconds, and on March 5, 1975, in Lyon, he reduced the time to 10 minutes, 32 seconds.
Later, Klein went on to extract a variety of roots: the 19th root of 133 digits (1 m. 43 sec.), the seventh root of 63 digits (8 m. 27 sec.), the 73d root of 500 digits (2 m. 9 sec.).
As explained in chapter 13, the difficulty of extracting integer roots of large numbers depends on the number of digits in the root - the size of the power is immaterial. The Guinness Book of World Records now accepts the extraction of the 13th root of a 100-digit number as a fair test; records now hinge upon improving the time required.
Klein has continually improved his times for extracting such roots. In Providence, Rhode Island, in September 1979, he achieved a time of 3 minutes 25 seconds; then in Paris, November 1979, 3 minutes 6 seconds; Leiden, March 1980, 2 minutes 45 seconds; London (BBC), May 1980, 2 minutes 9 seconds; Berlin, November 10, 1980, 2 minutes 8 seconds. Finally, on November 13, 1980, he got below two minutes - 1 minute 56 seconds. And on April 7, 1981, at the National Laboratory for High Energy Physics, Tsukuba, Japan, he established a new record of 1 minute and 28.8 seconds. With this he is fairly well satisfied. He plans now to attempt to split up the four- or five-digit numbers as the sum of four squares within one minute.
Klein describes much of his calculating as "semi-mental," in that he has the problem in view while solving it (thus obviating the necessity of memorizing it) and because he often writes down parts of the answer as he calculates them, before the entire answer has been found; this means that he does not need to keep the entire answer in mind before announcing it. For example, in multiplying he uses cross multiplication, which I allows him to write down the digits from right to left as they are obtained. After multiplying two eight-digit numbers for me "semi-mentally," Klein remarked, "Some people say to write down results as you go is not fair. You have to do it all in your head. But it takes five times as long and the audience will say, 'Forget it.'"
He asked me for two five-digit numbers and I gave him 57825 and 13489. In 44 seconds he multiplied these together mentally, without any intermediate results. He then repeated the experiment, writing down the answer as he calculated it - the time required was 14 seconds. "The first is more scientific, if you want. It is the real thing, but it's not what the people want."
Klein's passion, apart from numbers, is music. He is particularly fond of jazz and classical music. He says, "In New York every night I went to Jimmy Ryan's Jazz Club. I also did that when I was in New York two years ago. So when I popped in this time, they said, 'Hey, Flying Dutch-man, how are you? Have a drink. What shall we play for you?'" (Here Klein gave an excellent imitation of a trombone playing Ain't Misbehaving.)
"I play no instruments, pity enough. I've got about 600 LPs. Not very much, but.."
In spite of the justifiable pride Klein takes in his calculating ability, there is a passionate honesty in him as well. When I suggested that he may be the world's greatest mental calculator, he replied, "I'm not the world's greatest calculator. Perhaps the world's fastest calculator." In any case, Klein is surely one of the best mental calculators in history.