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 day-date 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 differen
t 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.
ILLUSTRATION CREDITS
Grateful acknowledgment is made for permission to reproduce the following:
Frontispiece: Detail from The Last Judgement (1536–1541) by Michelangelo Buonarroti. Sistine Chapel, Vatican Place, Vatican State. Courtesy of Alinari/Art Resource, New York.
1.1: The jaws of hell fastened by an angel, from the Psalter of Henry of Bloise, Bishop of Winchester, twelfth century. British Library, London, Great Britain. Courtesy of Bridgeman/Art Resource, New York.
1.2: Sinners in Hell, Last Judgement (circa 1240), anonymous. Relief. Mainz, Germany. Courtesy of Foto Marburg/Art Resource, New York.
1.3: Apocalypse of Saint John: Babylon Falls on the Demons (1363–1400) by Nicolas Bataille. Tapestry. Musée des Tapisseries, Angers, France. Courtesy of Giraudon/Art Resource, New York.
1.4: Detail from The Last Judgement (1432–1435) by Fra Angelico. Galleria d’Arte Moderna, Florence, Italy. Courtesy of Alinari/Art Resource, New York.
1.5: The Last Judgment by an anonymous Bologna artist, fourteenth century. Pinacoteca Nazionale, Bologna, Italy. Courtesy of Scala/Art Resource, New York.
1.6: Detail from The Last Judgement (1443) by Rogier van der Weyden. Altarpiece, center panel. Hôtel-Dieu, Beaune, France. Courtesy of Giraudon/Art Resource, New York.
1.7: The Opening of the Fifth and Sixth Seals, the Distribution of White Garments Among the Martyrs and the Fall of Stars (1498) by Albrecht Dürer. Woodcut from The Revelation of Saint John. Courtesy of Giraudon/Art Resource, New York.
2.1: Condemned in Hell (1499–1500) by Luca Signorelli. Fresco. Orvieto Cathedral, Chapel of San Brizio. Courtesy of Alinari/Art Resource, New York.
2.2: Detail from Condemned in Hell (1499–1500) by Luca Signorelli. Orvieto Cathedral, Chapel of San Brizio. Courtesy of Alinari/Art Resource, New York.
2.3: Detail from The Last Judgement (1536–1541) by Michelangelo Buonarroti. Sistine Chapel, Vatican Place, Vatican State. Courtesy of Alinari/Art Resource, New York.
3.1: The Vision of Saint John (1608–1614) by El Greco. Courtesy of the Metropolitan Museum of Art, New York. All rights reserved.
3.2: The Last Judgement (1808) by William Blake. Petworth House, Petworth, Sussex, Great Britain. Courtesy of National Trust/Art Resource, New York.
3.3: Los (1794) by William Blake, from The Book of Urizen. Courtesy of Library of Congress Rare Books & Special Collections/Rosenwald Collection, Washington, D.C.
3.4: Guernica (1937) by Pablo Picasso. Museo del Prado, Madrid, Spain. Copyright © 1997 Estate of Pablo Picasso/Artists Rights Society (ARS), New York. Courtesy of Giraudon/Art Resource, New York.
Stephen Jay Gould, Questioning the Millennium
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