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presents
As
Fast as Thought
A BBC Radio
programme
Recorded 6th October 1954
Broadcast 29th
October 1954
Hosted by Bertram
Bowden, with guests: Professor Alexander Aitken, Mr Wim Klein and Sir Cyril
Burt.
The following
narrative is based on notes I made while listening to the above programme. It
does not qualify as a transcript. That would require someone to note every word.
It does however include accurate descriptions of all the calculations, and a
complete transcript would probably not add anything of value or interest.
Italics indicate my own observations, made in retrospect.
RHF
Bowden opens the
programme by saying, “How very lucky we are to have as guests two men whose
abilities in mental calculation…” “…Professor Alexander Aitken and Mr
Wim Klein.”
As a foretaste,
Bowden asks Aitken and Klein for their thoughts on a number of his choosing. He
names 516, explaining that this was the number of the room at the BBC where the
programme had been discussed prior to its making.
Bowden turns to
Klein first. Klein mentions that 516 factorises as 2 squared times 3 times 43,
and that 43 is prime. Klein, not very audibly, says, “3 times n squared, 3
times 3 squared plus 4 squared.”
Klein appears
to be invoking some general rule for decomposing primes of a particular form
into a sum of squares, saying only the first part of this rule aloud. I am not
familiar with this rule (One of Ramanujan’s
perhaps?).
Klein adds, “3
squared times 4 squared times 5 squared.”
This product
is 3600. Possibly Klein meant to say PLUS instead of TIMES, as that would give
the sum 50, which coincides with the sum of the prime factors of 516 (2+2+3+43).
Klein then gives
the logarithm of 516 to 5 decimal places: “1.63347”.
A trivial
property.
Bowden then turns
to Aitken.
Aitken says, “I
notice some of the things Mr Klein notices, but not the logarithm. The first
thing I did was factor 516 as 12 times 43. I noticed also that 0.516 is
approximately 16/31.” Aitken then rattles off the decimal form of 16/31, “0.51612903…”.
Bowden then gives a
formal introduction of his two guests: “Professor Aitken is professor of
Mathematics at Edinburgh and a Fellow of the Royal Society. He has written many
books. He finds arithmetic a useful hobby, … Mr Klein is a
professional calculator who specialises in arithmetic. He studied medicine,
completing his studies before becoming a calculator.”
Bowden cites some
historical accounts of calculating prodigies, mentioning Buxton, Colburn,
Bidder, Gauss and Ampere. He remarks that he (Bowden) can understand athletic
ability and why some people are better athletes than others ¾
he contrasts himself with the runner Roger Bannister, but finds it difficult to
understand mental ability. He says that he wants to find out from Aitken and
Klein why they have such great mental ability and what methods they employ.
Bowden introduces
Sir Cyril Burt, a psychologist who specializes in giftedness. Bowden explains
the reasons for his own interest in calculating prodigies, saying that it
stemmed from an interest in calculating machines and the comparisons between
humans and machines. Bowden then notes some of the key differences between the
arithmetical processes of the machine and the human.
Bowden says that we
are now ready to test the two calculators. He explains that Mr Klein has a sheet
of paper in front of him and will write down the numbers as he goes. He also
explains that Klein finds it necessary to translate the question into Dutch,
sometimes via French (Klein had been living and performing in France recently)
and that this double translation often takes longer than the actual calculation.
Bowden says that Professor Aitken does his calculations entirely in his head.
Bowden explains that he (Bowden) has Barlow’s Tables in front of him to check
answers, and that the questions have been prepared beforehand. Bowden assures
the audience that neither calculator has any knowledge of the problems.
Bowden asks Klein
to “square 3039”. After 2 seconds Klein says “9 million, 235 thousand,
521”. It takes Klein about 5 seconds to say the answer.
Klein
apparently did not write down anything here.
Aitken agrees with
Klein’s answer, saying that he (Aitken) obtained the result simultaneously.
Aitken says that he will leave it to Mr Klein to explain how it was done.
Klein explains that
he split 3039 into 3000+39 ¾
“3 times 3 is 9, with 6 zeros, then 2 times 3 times 39 is 234, with three
zeros, 39 squared is 1521.”
Aitken says that he
“used a different method to Mr Klein”. Aitken then describes application of
a^2 = (a-b) x (a+b) + b^2, with a=3039 and b=39.
Bowden gives Klein
another number to square: 4531. Klein mutters for 2 seconds, and then gives his
answer: 20 million, 529 thousand, 961. The total time taken was again about 7
seconds.
Aitken says of
himself that he is not as fast as Mr Klein in such problems (squaring large
numbers) and mentions that he felt inclined rather to work out the square ROOT
of 4531. Bowden then asks Aitken to do just that. Aitken, clearly playing for
time, begins, “The square root of 4531, is, very approximately, … 67.31.”
Immediately, Klein says, “That is correct.” It took Aitken about 10 seconds
to give his four-digit answer.
Aitken
probably did this in two steps: First he recognised 4531 as 67 squared plus 42,
then he decimalised 21/67, which possibly he converted into 63/201.
Bowden mentions
that he had checked Klein’s ability before. He had pitted Klein against a
mechanical calculating machine. Klein was five or six times faster than the
machine/machine operator at 2 by 2, 3 by 3 and 4 by 4 multiplications. The
numbers were presented to the machine operator and to Klein at the same time.
To demonstrate
Klein’s skill at multiplying large numbers, Bowden gives Klein a problem to do
live. He gives Klein the first number, 12875, but before Bowden can say the
second number, Klein interrupts him with “that’s 5 cubed times 103”.
Bowden gives the second number, 9428. Klein mutters for about 10 seconds, then
gives the answer, “121 thousand, 385 million ¾
corrects himself ¾
121 million, 385 thousand, 500.”
Aitken says, “I
believe I would arrive at the same answer as Mr Klein, but by a different
method.” Aitken, without actually doing the calculation, explains that he
would, “multiply 9428 by 103, then divide by 8, but the answer is the same as
he (Klein) gets.”
Bowden contrasts
the methods of Aitken and Klein. He notes, “Klein’s methods appear to be
less imaginative, more routine and more pedestrian than Aitken’s methods,
though they take the same time.”
Bowden asks Aitken
and Klein if they would factorise some numbers for him. He gives Klein 4653.
After about 3 seconds Klein responds, “3 squared times 11 times 47.”
Aitken comments
that this is a bad example, since it is divisible by 3 and 1551 is 11 times 141,
and 141 is 3 times 47. In response to Aitken’s criticism, Bowden gives the
calculators a bigger number, 584321. Aitken says, “I’ll leave this one to Mr
Klein, as he’s more interested in the higher numbers.” After 5 seconds,
Klein says, “Ah! 31 times 16591” and adds that he does not know if 16591 is
prime or not. Klein then proceeds to test 16591 for factors. Klein can be heard:
“not 17, 19, … 31, 41. Ah! 47 times 353.” It took Klein about 15 seconds
to factorise 16591. Klein continues, “353 is prime and 17 times 353 is...”
Aitken interrupts, “6001!” Klein responds, “353 is 17 squared plus 8
squared.”
It appears
that Klein misheard Bowden, since he gives the factors of 514321 instead of
584321. This question was not one of the ones prepared beforehand, so the error
was not detected. It is curious that Klein tests 16591 for factors less than the
31 already identified.
Bowden says, “One
of Aitken’s specialities is decimalising fractions.” Bowden gives Aitken
23/47. Aitken repeats the problem, then reels off the digits
“0.48936170212765957446808510638” at a rate of 3 per second. Bowden feels it
necessary at this time to reassure the audience that neither calculator has been
told the numbers in advance. Aitken continues, “63829787234042553191 and that
completes the 46 digit period.” Astounded by this, Burt asks Aitken how he
knows that the digits repeat after 46 digits. Aitken reveals that he arrived at
“3191489” and “remembered 489 started off the expansion”.
Bowden asks Klein
if he is as fast as Aitken at decimalising. Klein says, “Oh no, much
longer.”
n/47 for all n
to 46 have a repeat period of 46 digits. For different n only the starting point
in the sequence of digits differs. Possibly Aitken had 1/47 memorized. Certainly
he would know that n/47 has a 46 digit period. Given Aitken's prompt reply and
the steady rate at which he gave the digits of the expansion, it is unlikely he converted 23/47 into, for example,
391/799, to enable short division by 8.
Bowden says that
Klein is an expert at doing calculations associated with the calendar and says,
“Sir Cyril has looked up some dates.” Burt asks Klein, “What day of the
week was it on October 13th 1842?” After 3 seconds Klein says,
“Thursday.” Burt confirms the accuracy of Klein’s answer, and then asks
Klein another, “What date was the first Saturday in March 1867?” Klein
mutters furiously (in French) for about 8 seconds, then announces, “2nd.”
Burt laughs in astonishment.
Bowden then
expresses his desire to test and compare the numerical recall of Aitken and
Klein. Bowden tells Klein and Aitken that he will call out a 10 digit number and
ask them to recall it in order. Starting with Klein, Bowden calls out,
“9465764078”. The digits are spoken at a steady rate of 3 per second. Klein
replies instantly, almost whispering, with the same number, spoken at the same
rate as it was presented.
Bowden asks Aitken
if he wants to try 10 or 15 digits. Aitken asks for 10. Bowden gives,
“9468398352”. Aitken replies instantly with the correct number, speaking at
a rate in excess of 5 digits per second. Despite his flawless performance,
Aitken commented that he was caught out by the similarity of the leading digits
of this number to those of the number given to Klein, believing that the same
number was being given to both himself and Klein.
Bowden mentions
Aitken’s “extraordinary ability to recite pi to 1000 places”. Bowden gives
Aitken a 5-digit sequence from “somewhere in the middle” (72458) and asks
Aitken to continue the sequence. Aitken gives the next 50 digits,
“700660631558817488152092096282925…” These digits are reeled off staccato
fashion, with no evidence whatever of any rhythm. Aitken explains that he
learned the digits by adopting a certain rhythm and demonstrates what this
rhythm is by tapping on the desk ¾
5 taps per second with about half a second between each of the 5 taps. Aitken
then recites the first 20 or so digits of pi in this rhythm, “14159 26535
89793 23846” Aitken says that he can memorize any number in groups of five
digits using this rhythm. Aitken compares memorizing numbers to learning music.
Bowden asks Klein
about his strategy for memorizing digit sequences. Klein says that he learns
numbers by heart rhythmically like Aitken, but always in groups of 3 digits, not
5.
Burt mentions that
the most interesting question from the point of view of a psychologist is that
of heredity, noting that some calculating prodigies share their gift with other
family members. Burt puts the question to Aitken and Klein.
Aitken says that
his daughter is quite a good arithmetician without practice. She can square
three digit numbers in a few seconds. Aitken believes that if he gave her a few
tips, then she would be very good indeed. Aitken said of his father, “He was a
very good arithmetician, adding halfpenny, pence, shillings, etc. but not
multiplication or division.” Aitken mentioned that his uncle¾his
father’s elder brother¾was
as good as himself at arithmetic, but that this may be only a tradition (myth),
and also that his grandfather was good at arithmetic but not exceptional. Aitken
stated that it was “always the eldest child who displayed the exceptional
skill”, adding, “I developed rather late.”
Klein says, “My
brother was very quick at multiplication, but was not interested. We used
different methods and we refused to learn each other’s methods.” Klein
mentioned that his father had a strong memory for general things, but not for
arithmetic.
Burt asks Aitken if
he thinks that his calculating ability is just a special application of an all
round ability, and also if there are any other members of his family who excel
at something, not necessarily arithmetic. Aitken believes his ability is a
special aptitude and not an application of an all round ability. Aitken suggests
that there may have been exceptional abilities in his family, but that his
recent ancestors were emigrants who did not have “higher schooling”.
Burt asks Aitken
for an example of memory other than for numbers. Aitken mentions history and
poetry, stressing that he does not know if it is exceptional. Aitken did remark
though that he found learning quite easy.
Burt states that a
characteristic of calculating prodigies is that they display their talents from
an early age. He cites as examples, Gauss, 3; Whately, 5; Ampere, 6; Countess of
Mansfield’s daughter, 5 or 6. Burt asks Aitken and Klein for the age they
began to calculate.
Aitken said that he
had no great interest until he was exposed to algebra, in particular the a^2-b^2
rule. Aitken realized that he could apply algebra to mental calculation. He
practised mental calculation from the age of 13.
Klein said of
himself that he started at age 7. His brother, who was a year his senior,
started at the same time. They began with numbers on motorcars. Klein said that
algebraic methods were not employed until later in grammar school. Klein
mentioned the a^2-b^2 rule.
Burt expresses his
desire to test the calculators’ numerical recall, with some more tests,
similar to those conducted earlier by Bowden. Burt states that for most people
the limit is eight digits. He proposes to test Aitken and Klein with numbers up
to 15 digits in length.
Burt tests Aitken
first. He says that he will give Aitken a 13 digit number. He calls out,
“3647382607291648” The digits are spoken at a steady rate of 2 per second.
Aitken replies immediately, “3647382607291, don’t know, 48”. Aitken
recited the number at a rate of about 4 digits per second, and certainly faster
than the rate of presentation. There was not the slightest delay between the
“1”, the “don’t know” and the “4”. Aitken complained that the
digits were presented at a rate that was “too slow” and that the rhythm was
“not that to which I’m accustomed”.
The number is
16 digits, not 13 as Burt claimed. Perhaps this was deliberate. Nobody picked up
on this fact. It is curious that Aitken erred after 13 digits ¾
the number of digits he was expecting.
For Klein, Burt
calls out the number “746931524735289”. Immediately Klein starts “746,
531, 753, 469”. Klein can be heard writing down the numbers while he is
recalling them. Burt says, “only about seven right.”
Burt asks Aitken
and Klein to recite the next numbers backwards. For Klein, Burt calls out,
“98253742” at a rate of 2.5 dps. Klein replies, at a rate slightly faster
than the presentation rate, “24735289”. For Aitken, Burt calls out,
“25871394” at 2.5 dps. Aitken replies “49317852” at about 3 dps. Aitken
commented that he “had to visualize this one because of the lack of rhythm”.
Burt quizzes Klein
on his number imagery, recalling that he had read somewhere that Klein confused
numbers that sound alike in Dutch, whereas Klein’s brother confused numbers
that looked similar. Klein confirmed this.
Burt proposed one
more memory test for Aitken. Three groups of three digits were to be called out.
Aitken was told to imagine these as the three different rows of a 3x3 matrix.
Aitken would then be requested to recall the digits in columns. Burt called out,
“517, 394, 682” at a rate of 1 row per second, i.e. 3 digits per second,
with a slight pause between rows. Aitken replied at the same rate as the
presentation, “536, 198, 742”. Aitken said this was visual. He also added
some comments on his use of visualization when recalling pi. Aitken said that he
had memorized pi as 20 groups of 50 digits and the beginning of each line
(group) comes into focus for just a moment – he had to “visualize the
joins”. He gave as an example the sequence “72458” proposed earlier.
Burt asked Aitken
and Klein if they predominantly heard or visualized numbers. Aitken said that he
could not judge ¾
“it was not seeing nor hearing. It was a compound faculty”. Klein said that
he heard the numbers in his own voice.
Burt remarked to
Klein, “Your lips move more than Professor Aitken’s.” Klein explained,
“I have to speak, and the more difficult the problem the louder I speak.”
Burt suggested that
there was a motor element to the imagery. Aitken said that was plausible and
added, “A strongly dynamic motor faculty ¾
not vision, not sound ¾
not static. Hidden, dynamic.” A dissatisfied Burt replied, “I feel that’s
vague.” Aitken uses violin playing as an analogy.
Burt reminds Aitken
that in the past he (Aitken) mentioned that he sees a page of music in his mind,
but not numbers. Aitken confirms this. Burt asks Aitken if, when he reads the
score of some opera, say figaro, whether he hears the whole orchestra. Aitken
replies that he hears “only the melody”. Aitken said that he would leave
that sort of thing to Sir Donald Tovey.
Burt asks Aitken if
he ever dreams about numbers. Aitken says, “I rarely dream numbers. Sometimes
algebra. Any arithmetic that is done in a dream is always correct.”
Burt is impressed.
Aitken says, “I have vividly coloured dreams on rare occasions. In dreams
numbers occur in some incidental way. If in a dream there is a train coming into
the station. I’ll see the number on the train and instinctively notice
properties of it ¾
its factors, for example.”
Burt asks Klein
whether he dreams numbers. Klein says, “no” and adds, “I never dream.”
Burt asks Aitken
about how much practise he gives to mental arithmetic. Aitken says, “I never
have opportunity or occasion to practise much. Twenty years ago I was better.
Multiplication, squaring and square root would daunt me. I suppose that’s just
ageing.” Burt puts it to Klein that he is improving in some types of
calculation, but is not as fast as when he was younger. Klein acknowledges this.
Burt asks Aitken if
a boy of average intelligence could achieve his (Aitken’s) level of ability.
Aitken says, “No. It must come from a gift, possibly latent. Even all the
practice in the world is no good without the gift.”
Burt asks Aitken
and Klein about the unconscious nature of their calculations. Aitken mentions
that if he sees a number on a tramcar or motorcar he might square it or notice
its factors, but prefers to switch this off. Klein says that he makes similar
observations in the street automatically – “839 prime. It’s no problem”.
Burt mentions that
for calculating prodigies, numbers are friends and that seeing a number is like
seeing a friend you met a long time ago. Burt asks Aitken if numbers have
“personalities”. Aitken replies, “Personality is a rather strong and
metaphysical word”. Aitken describes numbers as “an intrinsic field of
interest, which may be narrow, or may be broad, or may be rich”. Aitken
mentions that, “some numbers, like 811, have no properties at all, but others,
like 41, are deeply involved in many theorems, e.g. quadratic forms.”
Burt ends the
discussion by asking Aitken about the usefulness of his ability. Aitken replies
that its use is “limited".
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