Five Weeks

by Stephen Jay Gould

Poets, extolling the connectedness of all things, have said that the fall of a flower's petal must disturb a distant star. Let us all be thankful that universal integration is not so tight, for we would not even exist in a cosmos of such intricate binding.

Georges Cuvier, the greatest French naturalist of the early nineteenth century, argued that evolution could not occur because all parts of the body are too highly integrated. If one part changed, absolutely every other part would have to alter in a corresponding manner to produce a new but equally elegant configuration for some different mode of life. Since we cannot imagine such comprehensive change of every single part, each to the perfection of a new optimality, organisms cannot evolve.

Half of Cuvier's argument is undeniably sound. If evolution required such comprehensive alteration, such a process might well be impossible. But parts of bodies are largely modular and dissociable to a great extent. Alpha Centauri (not to mention more distant stars) didn't blink the slightest notice when little Susie pulled those petals off the daisy—"He loves me, he loves me not …" And even though the foot bone's connected to the ankle bone, evolution can change the number of stripes on a snail's shell without altering the number of teeth on its radula (jaw).

The functions of the brain, and human intelligence in general, also tend to be quite modular and dissociable. No g-factor, or unitary measure of "general intelligence," lurks within the brain, capable of ranking people according to their inherited quantity of a coherent thing, measured by a single number called IQ. (See my critique of this position in my earlier book The Mismeasure
of Man
.) Rather, intelligenceis a vernacular word that we apply to the large set of relatively independent mental attributes that build, in their entirety, something we call "mind."

The best, and classical, illustration of the relative independence of mental attributes lies in the stunning phenomenon illustrated by people who were once labeled with the stunningly insensitive name idiot savant—that is, globally retarded people with a highly precise, separable, and definable skill developed to a degree that would surprise us enough in a person of normal intelligence but that strikes us as simply miraculous in a person otherwise so limited. Some savants can do lightning calculation, multiplying and dividing long
strings of numbers instantaneously and with unfailing accuracy—but cannot make change from a dollar or even understand the concept. (Dustin Hoffman played such a character with great sensitivity in Rain Man.) Others can draw pictures, accurate to the finest detail, of complex scenes that they have viewed but once and for a fleeting moment—yet cannot read, write, or speak.

These people fascinate us for two very different reasons. We gasp because they are so unusual, and extremes always fascinate us (the biggest, the fiercest, the ugliest, the most brilliant). We need not be ashamed of this quintessentially human propensity. But savants also compel our attention because we feel that they may be able to teach us something important about the nature of normal intelligence-for we often learn most—about an average by understanding the reason for an extreme deviation.

We have favored two broad interpretations of these savants (each too simple, and probably both wrong, but still representing a reasonable first pass at formulating the problem). Do these people acquire their extraordinary skill because they discover one thing they can doand then work so very hard, and so assiduously, at developing it? In this case, any of us could probably master the savant's skill, but we would never choose to devote so much time to one mental operation. (In this alternative, the savant's brain does not differ from ours in the module devoted to his hypertrophied skill—and the phenomenon teaches us something about the nature of dedication.)

Or do these people develop their skill because deficiencies in one part of the brain's structure may be balanced by unusual development in another part? In this case, most of us could not learn the savant's skill even if we chose a path of single-minded devotion to such an activity. (In this alternative, the savant's brain may differ from ours in the module regulating his special skilland the study of this phenomenon may teach us something important about the physical nature of mentality.)

In any case, day-date calculation represents one of the most famous, and most frequent, of so-called "splinter skills" manifested by many savants. The subject has generated a great deal of study, well summarized in two recent books (Steven B. Smith, The Great Mental Calculators, Columbia University Press, 1983; and Darold A. Treffert, Extraordinary People: Understanding "Idiot Savants," Harper and Row, 1989). One obvious question has dominated the literature about mentally retarded and autistic day-date calculators: How do they do it?

The most obvious approach—simply asking a savant how he performs his day-date calculations-does not work. Few of us can give any decent explanation of how we accomplish the things we do best, for our truly unusual accomplishments seem automatic to us. (Sports heroes are famously unable to describe their extraordinary skills—"Well, urn, er, I just keep my eye on the ball and …" Intellectuals do no better in elucidating their literary or mathematical accomplishments—"Well, urn, I had a dream, and I saw these six snakes, and …") Savants, if they speak at all, will tend to say "I just do it"—and most of us could describe our special skills no better.

The literature has considered two basic modes—and results are typically inconclusive in illustrating the usual variety of reasons for human achievements. That is, some savants do it one way, others the other way, yet others in combination, and still others in a manner as yet undetermined. First, a savant might have extraordinary, even truly eidetic, skills in memorization. A daydate calculator might then simply memorize the calendars for a certain number of years and read the right day of the week for any date in any year directly out of memory. Second, a savant might develop an algorithm or rule of calculation, and then apply the rule so often, and with such concentration and dedication, that his calculation becomes extremely rapid and "second nature." At some point, the procedure may start to feel automatic.

Some savant day-date calculators do use memory alone—'and this method can be spotted because practitioners tend to memorize only a limited number of years. A savant who can do day-date calculation from, say, 1980 to 2020—but has no clue about dates in earlier or later years-has probably memorized forty years' worth of calendars (as researchers might be able to ascertain by checking a subject's bookshelf or asking if he owns a perpetual calendar for a limited number of years).

But many savant day-date calculators, including the young man described herein, use algorithms of their own invention. Some of these people, including my subject, can calculate effortlessly, and apparently instantaneously, sometimes across thousands of years, past or future, and with no apparent difference in the time needed to calculate a date two years or two hundred years from the present. The statement that some savants use algorithms still leaves two mysteries and complexities unaddressed—and these also figure prominently in literature on the subject. First, day-date calculation, as I showed in the last section, is a two-step process. You need, first, to know the day of the week for some reference year-usually the current year as given on a calendar. Then you can apply your algorithm to calculate the difference between your reference year and the year in question. Thus, no matter how good your algorithm, you still need to put some basic reference into memory. (Of course, you could begin any application by looking up the day of the week for this year on a portable calendar—but no self-respecting day-date calculator would use such a crutch.)

Second, and of most potential interest for insight into human mentality in general, the best algorithmic calculators, including my subject, do their reckoning far too quickly to be using their algorithm in an explicit manner. As a striking example, a graduate student studying George and Charles, the famous mathematical twins (and prodigious day-date calculators) so brilliantly and poignantly described by Oliver Sacks (in a chapter in The Man Who Mistook His Wife for a Hat), decided to try to equal their skills in day-date calculation by applying their method with the same singlemindedness manifested by many savants. He found that he could do the calculation, but he could not come close to their speed for a long time. Finally, and in a manner that he could never describe accurately, the technique just "clicked" and started to feel automatic. The student could then match the twins. Darold Treffert's book quotes a report by Dr. Bernard Rimland on this experiment:

Langdon practiced night and day, trying to develop a high degree of proficiency. … But despite an enormous amount of practice, he could not match the speed of the twins for quite a long time. Then suddenly, he discovered he could match their speed. Quite to Langdon's surprise, his brain had somehow automated the complex calculations; it had absorbed the table to be memorized so efficiently that now calendar calculating was second nature to him; he no longer had to consciously go through the various operations.

The young man I know, probably one of the best daydate calculators in the nation by now, is autistic and severely limited in cognition. His language skills are good, but his comprehension of intentionality and emotional causality is a virtual blank. He understands basic physical causality, and knows that a dropped object will fall to the ground, or a thrown ball hit the wall, but he cannot read human motivation or the "internal" reasons behind human actions. He cannot understand the simplest story in a book or movie. He can play a game in the sense of learning to follow the rules mechanically, but he has no idea why people engage in such activities and has never begun to grasp such concepts as scoring, winning, and losing.

Humans are storytelling creatures preeminently. We organize the world as a set of tales. How, then, can a person make any sense of his confusing environment if he cannot comprehend stories or surmise human intentions? In all the annals of human heroics, I find no theme more ennobling than the compensations that people struggle to discover and implement when life's misfortunes have deprived them of basic attributes of our common nature.

We tend to understand how the physically handicapped cope, but we rarely consider the similar struggles of the mentally handicapped. We must all order the "buzzing and blooming" confusion of the external world—and if we can't understand stories, we have to find some other way. This young man has struggled all his life to find regularities that might anchor and make sense of the surrounding cacophony. Many of his efforts have been dead ends and wild goose chases.

Since he reads faces so poorly, he struggled for years to find an additional clue in the pitch or loudness of voices. Does high mean happy? Does loud mean angry? He would play the same record at different speeds, converting Paul Robeson at 33 rpm to the sound of a woman's voice at 78 rpm—always hoping (or so I inferred) to induce some rule, some guide to action. He has never found it, though he still tries. When he was quite young, he developed some mathematical skills, and he put them to immediate use. He would time all his 33 rpm records, trying to find some rule that would correlate the type of music with the length of the recording. He got nowhere and eventually gave up.

Finally, he found his workable key-chronology. If you cannot understand stories, what might work next best as a general organizer? The linear sequence of time! You may not know why, or how, or whether, or what, but at least you can order all the items in a temporal series without worrying about their causal connections—this came before that, that before the other, the other before this-thing-here. He had triumphed. This young man can tell you something that happened on every individual day for the last twenty years of his life. Since he does not judge importance as we do, the event that he remembers often seems trivial to us, so we do not recall and therefore cannot verify his accuracy"—On that day, Michael Ianuzzi said 'Wow.' "But when we can check, he is never wrong—" On July 4, 1981, we saw fireworks on the Charles River."

I think I know why he first got interested in day-date calculation. Temporal sequencing had become the touchstone for his ordering of life. And what could be more riveting—and perhaps crucially important in some hidden way—than this interesting change in the weekdays of dates from year to year? There must be some rule behind all this. What could it be? So he struggled and found out. I watched his skills increase, but I never knew how he did the calculation.

If you are going to distinguish yourself by developing a narrow "splinter skill," I cannot imagine a more wonderfully useful choice than day-date calculation. Most people take an interest in knowing the day of the week on the date of their birth. But this information is not easy to come by. You can't look it up in the encyclopedia, and you can't find it on an ordinary calendar. Unless your mother remembered and told you the day, you probably don't know. To find out, you have to be able to perform day-date calculation—and most people can't.

This young man therefore becomes a priceless resource. I have seen him work a room like the best politician. He starts at one end and asks everyone the same question: "What day were you born on, and in what year?" His respondent says, "September 10, 1941," or whatever, and the young man replies without a second's hesitation, and in a special cadence well known to his friends and acquaintances—"A Wednesday." He is never wrong. A half hour later, I see him at the other end of the room. He has made the full circuit with all the aplomb of a diplomat—but with much more genuine interest generated. The feedback is also very gratifying for him—for people want to know and are genuinely grateful. They find his skill inscrutable and amazing—and they tell him so. A little stroking always goes a long way, especially for a man who has tried so hard to comprehend the confusion surrounding him, and has so often failed.

I always understood what this awesome skill in day-date calculation meant to him, but I yearned to find out how he did it—and he could never tell me. I figured out a few bits and pieces. I knew that he worked algorithmically, using this year's calendar (which he knows cold and apparently eidetically) as a reference and starting point. He knows the Gregorian rules for leap years and can therefore extend his calculations instantaneously across centuries and millennia. But what algorithm did he use?

He recognized both components of the general problem—algorithmic day-date calculators must, after all. He knew that the ordinary year contains fifty-two weeks and a day, and that days of the week therefore move forward by one for the same date in subsequent years—this year's Tuesday for any given date becoming next year's Wednesday. He also knew that an additional correction has to be made for leap years. But how did he put these two corrections together? What rule had he devised? I was stymied.

I then spoke to an English TV producer who had made a program on savants. He said to me: "Ask him if there is anything special about the number 28. All savant calculators that I have ever met have discovered this rule." But I didn't know the rule, so I asked him, "What's special about 28?" "Didn't you know?" he replied. "The calendar has a twenty-eight year repeat cycle. This year's calendar is exactly the same as the one for twenty-eight years ago."

Immediately I realized why this must be so—and I figured it out as any ordinary scientist with a modicum of basic mathematics would do. Of course. Two different cycles are operating simultaneously to cause the day-date shifts. First, a seven-year cycle based on the addition of a day each year-so that after seven years (disregarding leap years) the calendar comes back to where it began, and July 10 on a Wednesday becomes July 10 on a Wednesday again. Second, a four-year cycle based on adding an extra leap-year day every four years. So I dredged up an old calculational rule from my schooldays: If two cycles operate together, the multiple of their periods gives you the overall repeat time. Seven times four is twenty-eight. Thus, the calendar must work by a twenty-eight year repeat cycle-and this cycle becomes an obvious key for simplifying day-date calculations. You know the calendar for the current year already. The same calendar works for twenty-eight years ago. 1998 is the same as 1970. You already know that dates for 1999 will move one day of the week forward and 1971 is the same as 1999. And so it goes.

I had figured this out with some elementary arithmetic, but my autistic friend could not work this way. I was very eager to learn if he knew about the rule of 28. If so, would I finally grasp the key to his algorithm? Would I finally understand how he performed his uncanny lightning calculation? So I asked him: "Is there anything special about the number 28 when you figure out the day of the week for dates in different years?" And he gave me the most beautiful answer that I have ever heard—although I didn't understand a bit of it at
first. He said: "Yes … five weeks."

I was completely dumbfounded. Obviously, he had misunderstood, and his response had made no sense at all. So I asked again: "Is there anything special about the number 28 when you figure out the day of the week for dates in different years?" And he replied without hesitation: "Yes … five weeks."

I understood in a flash several hours later, and his solution was so beautiful that I started to cry. He could not use, or even understand, my arithmetical rule about multiplying the periods of two different cycles together. He could only work by counting concrete days, one after the other. He had figured out the following principle by thinking concretely in the only manner available to him: A year contains fifty—two weeks and some extra days—one extra day in an ordinary year, two extra days in a leap year. When the total number of extra days becomes evenly divisible by seven, then the calendar for that year is the same as the calendar I already know for this year. (The same argument works by subtracted days for past years, or by added days for future years.) If I can figure out a minimum span of years for which the number of added days is always exactly the same, and always exactly divisible by seven, then the calendar must repeat and I will have my rule.

So he began to count the number of added days concretely, one by one, year by year. Every span of years up to 28 couldn't work because the number of leap years varies. Thus, for example, a thirteen-year period may have four leap years (1960-1972) or three leap years (1961-1973). But when you reach 28 years—and never before—everything works out just right. Every 28-year span, whenever you start and wherever you finish, contains exactly seven leap years. (I am disregarding the Gregorian rule for omitting leap years at most century boundaries. As all day-date calculators know, this situation requires a special correction—and you must keep track of it separately.) Every 28-year span also includes exactly 28 extra days, arising from the rule that every year adds one day. Thus, every interval of 28 years adds exactly 35 days, no more, no less—one for each of the 28 years, plus seven additional days for the invariable
number of leap years. Since 35 is exactly divisible by 7, the calendar must repeat every 28 years.

I now finally understood how this consummate day-date calculator worked. He had added extra days concretely, the only mental method available to him. He could not use my mindless, memorized schoolboy rule—I still don't really know why it works—of multiplying the periods of coincident cycles together. He had added up extra days laboriously until he came to 28 years—;the first span that always adds exactly the same total number of extra days, with the sum of extra days exactly divisible by seven. Every 28 years includes 35 extra days, and 35 extra days makes five weeks. You see, he had given me the right answer to my question—but I had not understood him at first. I had asked: "Is there anything special about the number 28 when you figure out the day of the week for dates in different years?" and he had answered: "Yes … five weeks."

May we all make such excellent use of our special skills, whatever and however limited they may be, as we pursue the most noble of all our mental activities in trying to make sense of this wonderful world, and the small part we must play in the history of life. Actually, I didn't quote his beautiful answer fully. He said to me: "Yes, Daddy, five weeks." His name is Jesse. He is my
firstborn son, and I am very proud of him.

This essay was reprinted from Stephen Jay Gould's book Questioning the Millennium and was reprinted with permission of the publisher.

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is a literary historian and author of Not Even Wrong: Adventures in Autism.

is an artist and author of Different Like Me: My Book of Autism Heroes.