Dead Reckonings

Mental calculators of yesteryear were usually described in magazines, newspapers and books in ways that can be startling in our more cynical age. But even today newspaper articles, documentaries and television features on modern lightning calculators appear almost regularly, often with a “hook” such as diminished capabilities in other areas (the “Einstein” effect). Surely there must be some reports that try to be objective, but I haven’t found them. At best they are naively written by people with little mathematical background; at worst they use considerable license (deception, really, if only by omission) to present a better story. This part of the essay is not directly related to the historical art of mental calculation itself, but I think it serves as a cautionary tale in evaluating articles on it.

Hans the Clever Horse

Let’s take a quick look at a historical example of media misrepresentation, in this case an unintentional one. In the late 1800s and early 1900s the horse shown in the pictures here (Clever Hans) was thought to have the ability to perform arithmetic as well as other reasoning tasks expressed by tapping a hoof a certain number of times. The New York Times wrote a feature on the horse in 1904 (BERLIN’S WONDERFUL HORSE; He Can Do Almost Everything but Talk—How He Was Taught) that brought enough attention to the matter that the German board of education created a team of experts to investigate the situation. Following extensive testing, the New York Times reported (correctly) that the committee had found no evidence of trickery and concluded the horse was exhibiting genuine skills. Only later did the psychologist Oskar Pfungst conduct enough blind tests to determine that the horse was reacting to unconscious cues by the questioners. (For a really good read on this, see here.)

So you can’t always believe what you see or read, and more generally it takes a critical look (and maybe some cynicism) to separate the chaff from the wheat. And when the story is worth retelling, and particularly when the calculator is a savant, it’s often difficult to be objective about the subject.

Innocent Sources of Hyperbole

Often information for newspaper articles is taken from promotional material or verbal descriptions by biased acquaintances or naïve observers, and of course there is always the temptation to embellish the truth a bit. There are accounts I’ve read of confederates in the audience asking questions or seeding problems with numbers having special properties that make, say, multiplication or division with another genuinely produced number much easier. This isn’t unlikely if you think of the “showman” type of lightning calculators, ones who mix these demonstrations with mentalism or magic (Arthur Benjamin, however, is a true lightning calculator as well as a magician). I can also say that the few times I have seen a mental calculator in action, the audience cannot distinguish between presentations of pure mental calculation and simple, standard ways of completing a magic square, for example, that I think of as dross. I also think it’s fair to say that incorrect answers are seldom reported, particularly if the calculator corrects the error.

Sometimes the problems posed by honest people turn out to be simple for the calculator given their extensive practice. For example, Smith records a number of questions that involve the number of seconds in some number of years; hours in some months, days and hours; cubic yards in some cubic miles; and so on for various simple multiples of common unit conversions known by any calculator of the time.

It also happens that the calculator gets lucky in a problem selection or in an answer, and then it’s one for the record books. Although I’m not a lightning calculator by any stretch, I’ve certainly benefited from a lucky guess. There was a particularly complicated calculation in my physics class once, involving many terms with powers in the numerator and denominator. As I was wont to do while students reached for their calculators, I wrote down what I thought the first few digits were, which was actually a stretch for me given the problem, and then just wrote two more digits randomly. When the first student read out the answer from the calculator only the last digit was off, and only by 1. There was absolute silence in that classroom as I turned and changed that last digit, and of course I never said a word about it.

People sometimes stumble by providing a problem they think is difficult, such as choosing odd numbers or even prime numbers, without realizing that the particular numbers offer a convenient shortcut or, more likely, that the calculator has already memorized the results for these numbers. We will see later that in a documentary on Tammet the researchers decided to ask him a high power of a 2-digit number. Were they going to pick an even number, or maybe one in which the digits were the same? Not likely—primes seem ideal, and there are limited numbers of them. In fact the limits of the calculator display and the powers they selected limited the number to less than 40, so how many likely numbers are there? (he was asked for 37⁴, 27⁷ and 31⁶ in the documentary, and it is true that 27 is not prime). I can’t claim that Tammet knew these, but he may have at least known some intermediate powers of these that might have helped (and Tammet has prodigious powers of memory for numbers). This is fine and fair game for any lightning calculator in my opinion. Klein, for example, knew a wealth of number facts, such as “the first 32 powers of 2, the first 20 powers of 3, and so on.” In fact, in referring to Dase’s calculational efforts, Gauss wrote

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One must distinguish two things here; an extraordinary memory for numbers and true calculating ability. These are, in fact, two completely separate quantities that may be connected, but are not always.

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And it’s easy to read through an account and unthinkingly accept the writer’s assumptions. When I was young I read an account in which the interviewer wrote down a 20-digit number on a napkin and presented it to a memory expert for 15 or 30 seconds, after which the person could read it backwards and forwards. I realized that I could certainly do that, and a lot of people can, say by mnemonics or by grouping it into five 4-digit numbers. In tests by Binet, Diamandi was able to memorize on average 11 digits in 3 sec, 16 digits in 5 sec, and 17 digits in 6 sec, although Binet indicates a significant error rate. Is this so tough? After all, 3 seconds is really a longer span of time than you might think. At Eberstark’s request, Smith tested him by reading aloud single digits at a tempo specified by the calculator, about 1.75 sec between digits, for 20 digits, (Eberstark at the end extended this to 40 digits). Is this hard? There are those who do in fact perform amazing feats of quick memorization (in tests Salo Finkelstein repeated a 20-digit and 25-digit number after exposure for 1 sec apiece, a 33-digit number exposed for 2 sec, and 39 digits exposed for 4 sec), but the lesson here is to be critical when reading articles or watching programs.

As a final example, in a 2005 performance in the second Arthur Benjamin video listed earlier and found here, four audience members were brought on stage at the start to verify his answers on calculators. Benjamin did a variety of calculations, most of them correctly, but it’s interesting that despite his turning to them to request verification, two of his five answers in squaring 3-digit numbers were incorrect. But none of the four challenged his answers, and to be honest, I wouldn’t have had enough confidence that I entered the digits correctly to have held up a show like that either. So don’t trust observers or judges.

Conscious Bias in Reporting

Sometimes the writer or director purposely skews the reporting in ways that are probably conscious but not that serious—sins of omission and that sort of thing in a light piece. For those with an interest in the subject beyond casual reading, it’s important to notice these nuances.

Smith’s book is rife with contemporary accounts that use phrases such as “in an instant” or “in a flash” or “in the blink of an eye” and so forth. And the details of the task are seldom presented, even in structured tests by researchers, a fact that absolutely amazes me. Did the calculator repeat the problem back to the questioner? Did the questioner write the question down in front of the calculator, did the calculator have the problem in view during the test, and was timing (if there was any) stopped when the first digit of the answer was being written or the last, or when the calculator said “Done” or when the last digit was recited? And we have seen that there are particular types of problems (e.g., the number of seconds in a given number of years) that benefit hugely from memorized facts. Smith also points out instances in which the set of test questions by researchers all shared the same shortcut property—why is that? How many questions were asked in all? Were just the correct, speedy ones reported? One rarely if ever has these facts in an account of a lightning calculator.

This continues today, of course. Let’s look at a common example of subtly slanted reporting. You might think I’m seeing bias where there is none, but when you read enough of these accounts you begin to see patterns. Alexis Lemaire is an extremely talented mental calculator with a specialty of extracting 13^th roots of 200-digit numbers, a feat that I don’t believe is attempted by others. So my comments here are only on the reporting and in no way reflect on Mr. Lemaire or his abilities.

So here’s a typical news report of a record time set by Lemaire at the Oxford Museum of the History of Science, dated July 30, 2007, by BBC News and found here.

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The task is to find the 13th root of 85,877,066,894,718,045,602,549,144,850,158,599,202,771,247,748,960,878,023,151,

390,314,284,284,465,842,798,373,290,242,826,571,823,153,045,030,300,932,591,615,

405,929,429,773,640,895,967,991,430,381,763,526,613,357,308,674,592,650,724,521,

841,103,664,923,661,204,223.

The answer’s 2396232838850303. Multiply that by itself 13 times and you get the above. Even with a calculator you wouldn’t beat Alexis Lemaire doing the calculation in his head.

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Another article from 2005 on such a record, shown in the figure above, can be found here.

Now let’s look at this article of another such record from any of the seven reports I found very quickly online via Google. For direct comparison with the first one above I’ll choose BBC News again, dated December 11, 2007, and found here.

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The fastest human calculator has broken his own mental arithmetic world record.

Alexis Lemaire used brain power alone to work out the answer to the 13^th root of a random 200-digit number in 70.2 seconds at London’s Science Museum.

The 27-year-old student correctly calculated an answer of 2,407,899,893,032,210, beating his record of 72.4 seconds, set in 2004.

The so-called ‘mathlete’ used a computer package to randomly generate a number before typing in the answer.

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So here we are given a little more information on how the test was performed. But do you see the real difference here? The randomly generated number is not reported, just the root. Why is that—generally the huge number is much more impressive to present than the root. Well, let’s see what that randomly generated number was:

91474397281474512894803677416201430283564210503432385339561327276933454229
60930464647192509451811477101625889659290744142634989755650414557096020392
5503679105245199142338806082494254050610000000000000

It doesn’t look so random. In fact, it wasn’t the power that was randomly generated (after all, what are the odds that a randomly generated number would be a 13^th power?), but rather the root was randomly generated and the power calculated from that value. Which is fine, but it seems apparent to me, at least, that they would have reported the power if it didn’t end in thirteen zeros. The reader might immediately intuit that last digit of the root is 0, so it detracts slightly from the effect and makes them consider that it was a lucky break. In fact the last digit is always identical in a number and its 13^th power so it’s always trivial to find it, but this now makes the second-to-last digit trivial (the digit 1). I see a little bias on the part of the reporter, and I saw this in every report of this event I could find.

Let’s take a news report of another record-breaking event from November 16, 2007, found here.

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27-year-old Alexis Lemaire from France has set a new world record by mentally calculating the 13th root of a 200-digit number in 72.4 seconds. He correctly identified the answer as 2,397,207,667,966,701. The previous record was 77 seconds.

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No 13^th power listed here either. And I believe the reason is that this power is

86332348800352843610126990022313468510477370930755992152681390347795323097
51168717005763648080727141383324712170576311110855841562345802001852561285
2897226196105357173387251523920946707380414694987101

With the power and root in view it doesn’t take too long to figure out that any power ending in 01 would have roots ending in 01, so this was again a case in which the last two digits are found instantly. This may not have been a lot of help to Lemaire, as he probably knows all two digit endings of 13^th roots, but again this is all about the reporting. Also, these are situations where it’s easy for us to see the advantages of a particular number, whereas lightning calculators have a wealth of stored number facts that can make certain problems much easier in a less apparent way, and it only takes one lucky number to break a record.

Deconstructing the BrainMan Documentary

Let’s take a look at a documentary that in my opinion inadvertently reveals itself as duplicitous. This may seem a bit tedious, but I’m kind of proud of the mathematical detective work I did in uncovering this.

Daniel Tammet currently holds the European/British record for memorizing pi (22,514 digits in all!) as listed here, he has appeared on 60 Minutes and David Letterman’s show, and he has written a recent autobiography titled Born on a Blue Day: Inside the Extraordinary Mind of an Autistic Savant. He has some talent with mental calculation as well, and he was featured in a popular 2004 documentary called BrainMan (titled The Boy With The Incredible Brain in the UK). The documentary won a Royal Television Society award in December, 2005, and was nominated for a BAFTA in 2006.

Tammet experiences synaesthesia, the ability to see or experience numbers as shapes, colors and textures. Here’s a typical excerpt from an article on him:

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Tammet is calculating 377 multiplied by 795. Actually, he isn’t “calculating”: there is nothing conscious about what he is doing. He arrives at the answer instantly. Since his epileptic fit, he has been able to see numbers as shapes, colours and textures. The number two, for instance, is a motion, and five is a clap of thunder. “When I multiply numbers together, I see two shapes. The image starts to change and evolve, and a third shape emerges. That’s the answer. It’s mental imagery. It’s like maths without having to think.

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Now I was asked awhile back about Tammet’s solution of 13/97 in the BrainMan documentary. I had not seen it, but I replied that division by a two-digit number like 97 is not difficult (130/97=1 remainder 33, 330/97=3 remainder 30+3(3)=39, 390/97=4 remainder 2, etc., so we get .134… and so on—lightning calculators can fly through this). However, we saw earlier here than the reciprocal of 97 consists of repeating groups of 96 digits. I thought it likely that either half the repeating group or all the repeating group of 1/97 was memorized, because changing the numerator to any 2-digit number less than 97 simply cycles the starting position of the repeating group to another location. In fact, Aitken had remarked on the commonly proposed problem of this reciprocal in his talk:

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Here the remark was made that memory and calculation were sometimes almost indistinguishable to the calculator. This was illustrated by the recitation of the 96 digits of the recurring period of the decimal for 1/97, checked by Dr. Taylor. Probably because 97 was the largest prime number less than 100, this particular example had been frequently proposed.

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Actually, I suspect division by 97 is often asked because it takes a whole 96 digits before the digits start repeating.

Later that day it occurred to me that there might be a way to detect whether that problem was solved by Tammet through a memorized repeating group. The reciprocal of 97 is

1/97 =

.010309278350515463917525773195876288659793814432|

98969072164948453608244226804123711340206185567|

010309278350515463917525773195876288659793814432|

98969072164948453608244226804123711340206185567|

01030927…

and also

13/97 =

.134020618556701030927835051546391752577319587628|

865979381443298969072164948453608247422680412371|

134020618556701030927835051546391752577319587628|

865979381443298969072164948 453608247422680412371|

13402062…

I’ve added vertical bars at each half of the 96-digit repeating group. We can see that each decimal expansion starts repeating every 96 digits. In addition, each digit in one half of the repeating group is the difference from 9 of the corresponding digit in the other half of the repeating group as mentioned earlier in the Fast Division section of this essay. So to produce 1/97 we can just memorize the first 48 digits of the repeating group, and then repeat that it but subtract each digit from 9. Again, Aitken had done that in anticipation of being asked for it in performances.

You can see that 13/97 has simply cycled the repeating group to the position near the end that starts with 134. The starting point for a given numerator is not predictable, but we can just divide 13 by 97 to a few digits (.134) to find the start, or we can multiply 13 by the first few digits of the repeating group for 1/97 (.0103) to find 134 as the starting point. If only the first half of the repeating group of 1/97 is memorized, we would see if 134 is in that group, and if it isn’t there (like now) we look for the 9’s complement 865 in that group, which is found near the end. So we start at that point, listing the 9’s complement of each digit as we cycle around the start of that half-group, and when we reach 865 again we repeat the steps but don’t take the 9’s complement. So we can always get away with memorizing just 48 digits to divide any number by 97. If the numerator is greater than 97, it’s just a whole number and a fraction with a numerator less than 97, so we end up being able to divide any number at all by 97 this way. (Technically speaking, we only need to memorize 47 digits because the repeating group for division by any number ending in 7 ends in 7, but then we’d have to remember that fact.).

Well, I thought that given the different starting position of 13/97, if Tammet were using a memorized half or full repeating group of 1/97, a verbal hesitation might be detectable at the end of this group as he “resets” his memorized or mnemonic digits to the start of this group. The 1/97 repeating group ends with the digits 567 and then cycles back to the beginning to 010… This 567 sequence occurs in the 11th to 13th position in 13/97. If Tammet hesitated between stating the 7 in the 13th position and the 0 in the 14th position in reciting 13/97, it would be evidence that the memorized group (or half-group) of 1/97 was being utilized.

So after all of this I finally watched the documentary online. You can find the complete UK version of the documentary in 5 parts on YouTube. The first part is found here. (Google Video has the U.S. version in a single video, but it does not include the scene with the whiteboard that I will refer to below.)

The 13/97 calculation is not far into the documentary (at 1:49 in Part 1), so it’s quite interesting to watch all of it up to that point to see how the shapes/crystallization process is described. He actually does the 13/97 calculation twice back-to-back, it turns out.

We don’t hear the problem posed to Tammet in the video, so we don’t know the delay between when the question was asked and the response was begun. Tammet slowly recites the digits of the answer all the way up to 5..6..7 and then—he comes to a total halt! He has lost his place, the questioner mentions that he is carrying on, he says he’s carrying on, then he says to tell him to stop or … and the interviewer stops him. I about fell off my chair. The interviewer asks him how many places he can do it to, and Tammet replies, “A hundred—nearly a hundred.” To me this reveals that he has done division by 97 before, and as we know if you can do 96 digits you can repeat them as long as you want to.

And this brings me to something very misleading—that the interviewer implies here that division gets harder as you get further into the solution. This is absolutely false—in division it’s just as easy to get the 1,000th digit as it is to get the 2nd digit. It is not like, say, a square root.

And I’m in luck—Tammet is going to repeat his answer after the interviewer retrieves a computer to get more digits. And here I will say that anyone who digs up an old 8-digit calculator to go test a savant on his calculating abilities, especially on division, can only be setting up a dramatic scene in a fluff piece. And another thing: the interviewer tells Tammet to wait, they are going to go find a computer, then come back, boot it up, sync the camera to the vertical sync of the video card, run the calculator application, and ask him the very same problem to see how many digits he can do?? And what would someone like me be doing the whole time they’re fiddling around? I’d be furiously calculating more digits in my head, that’s what.

But Tammet starts reciting again at about the same rate as the video scans the digits on the computer screen. He gets to the 5..6..7… and to my astonishment at that exact moment (and I mean exactly at the instant the next digit would be uttered) a voiceover is spliced into the video, saying two superfluous things—that every digit is correct (which we would know if the voiceover wasn’t there) and that he will eventually exceed the 32 digits of the computer (which we find out when we get there). Also, at this same point a video cut is shown of his hands making movements on the table. Then the audio and video return to him reciting about a dozen places later. If you replay the video and you continue reciting the correct digits during the voiceover, you find that Tammet would have had to have sped up significantly to have been at that point when they return to his recitation, although to be fair he does speed up at the very end of it all.

So it strongly appears to me that the documentary covered up for him on what I think must have been some sort of difficulty at the point I predicted. Not a big deal, maybe, but the producers went to a lot of trouble to deceive us on this, and that’s makes me question the validity of the whole enterprise.

In Tammet’s book on page 4 he says he “calculates” divisions like 13/97 by seeing spirals rotating in loops that seem to warp and curve, and in fact if you go to nearly the end of Part 4 of the documentary (from 9:41 to 9:44 here) and look carefully when Tammet is tracing such a spiral on the whiteboard (see the frame capture here), you’ll find it is being drawn right below “1/97”. So this divisor pops up again, lending credence to the theory that it was half- or fully-memorized.

Finally, let’s look at the three integer powers that are asked of him. Very early in Part 1 (at 0:58 in here) the narrator says that Tammet was asked to find 37⁴. We don’t see it asked, we just see the interviewer punching 37 x 37 x 37 x 37 extremely slowly into the calculator, followed by a continuous pan to Tammet, who looks up and recites the answer. So the question was asked at some unknown time prior to the entry into the calculator.

Later in the Part 4 of the documentary (at 0:40 in here) Tammet is asked to find 27⁷ and 31⁶ in two apparently unplanned, poorly executed tests by two neuroscientists (are there no x^y keys on these calculators??). We’ll never know how long it took to find the results because the documentary has so many cuts injected there that our sense of time is destroyed while the background music gives a false sense of continuity. We do see that they don’t start a small timer until 4 seconds after one of the problems is given. Again, the producers of the documentary mislead the audience by compressing the timescale. And for those who still might have thought the documentary to be unbiased, a voiceover appears during the latter calculation to blame whatever delays there were (what were they?) on jet lag.

So to summarize all this, in the process of trying to analyze Tammet’s method I found strong evidence that the BrainMan documentary in several ways actively misled the viewers. And of course this all has to do with the producers of the documentary themselves, not Daniel Tammet. And that’s why you have to be critical of these sorts of things.

Now let’s consider the first part of the documentary listed earlier on Rüdiger Gamm found here. Very early into it, just after 0:40 sec, Gamm announces to an auditorium that he will attempt to divide the prime number 109 into a 2-digit number provided by an audience member. He will attempt to go 100 digits after the decimal place. After receiving a number of 93, Gamm repeats the problem “93/109” and focuses on the problem for a total of 11 sec. Then he starts reciting the digits, very soon accelerating and reciting the digits as fast as he can say them.

Every alarm in your head should be going off about now. Is the number Gamm chose (109) one of those primes whose reciprocal has the maximum possible repeating group (108 digits)? Did he recite only 100 digits so the repetition after 108 digits wouldn’t be noticed? Yes, and in my opinion, yes. Here’s the reciprocal of 109 with vertical bars separating halves of the repeating group as in our earlier example for 97:

1/109 =

.009174311926605504587155963302752293577981651376146788|

990825688073394495412844036697247706422018348623853211|

009174311926605504587155963302752293577981651376146788|

990825688073394495412844036697247706422018348623853211|

0091743119…

So the repeating group does have 108 digits, and as always in this case, the digits in the second half are the 9-complements of the corresponding ones in the first half. So let’s take the submitted numerator 93 and mentally divide it by 109 to three digits. We can divide 93 by 110 instead and adjust for the offset in each step by adding the previous digit, as presented earlier in the Fast Division section for division by a number ending in 9:

93/11 = 8 remainder 5

(50+8)/11 = 5 remainder 3

(30+5)/11 = 3 remainder 2

and we locate 853; it’s in the second half of the group near the end. If the repeating group for 109 is memorized (or even half of it as described earlier since the other half is the 9’s-complement), it’s child’s play to recite the digits.

Now I don’t know how Gamm actually performed this feat. If you practice just a bit with the adjusted division process you can develop a kind of rhythmic cadence as you go:

          93   8   58   5   35   3   23   2   12   1   11   0   10   0   100   9   19   1   81   7   …

This is remarkably easy if you try it without reading it. Stating the bolded digits out loud really helps to append them to the remainder of the next division. Gamm does seem to develop a sort of cadence in the video, and he is a phenomenal calculator, so it’s likely that he is just amazingly fast at this. In any event, to judge the performance it’s important to realize that an adjusted division technique exists, and it’s also worth noting that with some memorization you too could walk into an auditorium and perform as well as Gamm on this.

The Appeal of the Mental Calculator

The study of lightning calculators of the past is a fascinating one for me from a mathematical aspect more than a psychological one. We’ve seen years of articles by educators bemoaning the dependence of students on calculators, but I see little in school textbooks on mental math other than simple estimation. And yet when I have presented basic methods of mental calculation to classes (elementary and college), I’ve met with incredible interest. Certainly the BrainMan documentary is a very popular one. But these types of presentation generally ascribe abilities in these areas to mysterious machinations in the minds of remote geniuses, which makes for a good story but can be discouraging. In fact, these individuals through talent and training acquired a knack for racing headlong through calculations that are not mysterious at all once the methods are taught.

And they are not being taught. Mental calculation can be a highly creative and satisfying endeavor offering a variety of interesting strategies, more than I have presented here and many more than most people realize. It is a skill that engages both children and adults, and one that naturally leads to a real familiarity with the properties and relationships of numbers. It provides a useful and fun approach for developing a number sense and generating a true appreciation for the elegance of elementary mathematics. It should not be a neglected art.

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