Volvamos atrás en el tiempo y situémonos en una cierta escuela, en el año 1787.

En ella había un maestro que era un bruto llamado Büttner y digo bruto porque afirmaba que su idea de educar a los niños era llevarlos a un estado de aterrada estupidez tan grande como para que olvidaran su nombre. Todo un pedagogo. Dicho maestro propuso como ejercicio sumar todos los enteros consecutivos del 1 al 100. El primero en acabar el ejercicio debía dejar su pizarra sobre la mesa del maestro, el siguiente alumno encima de la del primero y así sucesivamente. Con ello pensó que tendría una hora ocupada la clase, pero tras unos pocos segundos, uno de los alumnos, un chaval de 10 años, se levantó, puso su pizarra en la mesa del profesor y se fue a su sitio. Esperó una hora a que finalizaran sus compañeros. Mientras Büttner miraba las pizarras con resultados incorrectos iba calentando su bastón para el primer chaval. Pero para su sorpresa vio que la pizarra estaba con la respuesta correcta: 5050. Le preguntó cómo lo había hecho.

Le dio la siguiente explicación: imaginó que escribía la suma dos veces, una al derecho y otra al revés una encima de la otra

  1 +  2 +  3 + ... + 98 + 99 + 100
100 + 99 + 98 + ... +  3 +  2 +   1

Si sumamos columna a columna vemos que todas dan lo mismo: 1+100=101, 2+99=101, 3+98=101, etc. Así que la respuesta es 100 veces 101 dividido entre 2 ya que hemos sumado la serie dos veces. No está mal para esa argumentación para un chaval de 10 años, ¿no?.

A partir de ahí, Büttner siempre trabajó con el chaval atiborrándole de libros de texto, cosa que este último le agradeció toda su vida.

El nombre de este chaval: Karl Friederich Gauss.

Después de oír o leer el apellido de este nombre le viene a uno a la cabeza la distribución de errores que hoy se conoce como curva o campana de Gauss o distribución Normal.

Nacido en Braunschweig en 1777 fue un niño prodigio y continuó siendo un hombre brillante toda su vida. Aprendió a calcular antes que leer. A la edad de 3 años ya corregía las sumas que hacía su padre e impidió con ello que pagara de más a sus empleados, dado que encontró un error en sus libros de contabilidad.

Shortly after his seventh birthday Gauss entered his first school, a squalid relic of the Middle Ages run by a virile brute, one Büttner, whose idea of teaching the hundred or so boys in his charge was to thrash them into such a state of terrified stupidity that they forgot their own names. More of the good old days for which sentimental reactionaries long. It was in this hell-hole that Gauss found his fortune.

Nothing extraordinary happened during the first two years. Then, in his tenth year, Gauss was admitted to the class in arithmetic. As it was the beginning class none of the boys had ever heard of an arithmetic progression. It was easy then for the heroic Büttner to give out a long problem in addition whose answer he could find by a formula in a few seconds. The problem was of the following sort, 81297 + 81495 + 81693 + ... + 100899, where the step from one number to the next is the same all along (here 198), and a given number of terms (here 100) are to be added.

It was the custom of the school for the boy who first got the answer to lay his slate on the table; the next laid his slate on top of the first, and so on. Büttner had barely finished stating the problem when Gauss flung his slate on the table: "There it lies," he said—"Ligget se'" in his peasant dialect. Then, for the ensuing hour, while the other boys toiled, he sat with his hands folded, favored now and then by a sarcastic glance from Büttner, who imagined the youngest pupil in the class was just another blockhead. At the end of the period Büttner looked over the slates. On Gauss' slate there appeared but a single number. To the end of his days Gauss loved to tell how the one number he had written down was the correct answer and how all the others were wrong. Gauss had not been shown the trick for doing such problems rapidly. It is very ordinary once it is known, but for a boy of ten to find it instantaneously by himself is not so ordinary.

1784 wird der kleine Johann, wie er damals noch hiess—später nannte er sich Carl Friedrich, unter Weglassung des ersten und Umstellung der beiden anderen Bornamen—, in der Katharinenvolkschule in Braunschweig eingeschult. Lehrer Büttner wirkte hier in einer "dumpfen niedrigen Schulestube, mit einen unebnen abgelaufenen Fussboden, von der man nach der einen Seite auf die beiden schlanken gotischen Türme der Katharinenkirche, nach der anderen auf Ställe und armselige Hintergebäude hinausblickte". Wo andere Mittel der Erziehungskunst versagten, griff der Stock ein, den Lehrer Büttner stets mit sich führte, wenn er durch die Klasse schritt. Nach zwei Jahren Anfangsunterricht trat der kleine Johann Gauß mit anderen begabten Kindern in die Rechenklasse des Lehrers Büttner über. Es wurde da an Ort und Stelle auf der Schiefertafel fleißig gerechnet. Büttner stellte Ausgaben. Wer fertig war, legte seine Tafel auf einen großen Tisch im Klassenzimmer, die beschriebene Seite zuunterst. Darauf legten ebenso die anderen fertig werdenden Kinder die Tafeln mit ihren Ergebnissen. Dann begann das Gericht über die Könner und Nichtskönner. Es begab sich, daß Lehrer Büttner einmal für längere Zeit Ruhe vor der Rechenklasse haben wollte und dachte, die werde er haben, wenn er ausgebe, die Zahlen von eins bis hundert zusammenzuzählen. Dazu hatte er ja geringe Mühe, die Richtigkeit des Ergebnisses nachzuprüfen, weil ja bekanntlich 5050 herauskommt. Doch kaum hatte er die Ausgabe gestellt, da legte der kaum neunjährige Gauß seine Tafel mit dem richtigen Ergebnis auf den Tisch mit den plattdeutchen Worten: "Ligget se!" So berichtete er oft selber im Laufe seines Lebens. Das Kind hatte seine erste selbständige Entdeckung gemacht. Es hatte die Summenformel der arithmetischen Reihe für sich entdeckt. Lehrer Büttner kannte sie natürlich auch. Aber Gauß hatte bemerkt, daß man ja nur zur ersten Zahl die letzte 100, zur zweiten die vorletzte 99 usw. zuzuzählen braucht, um immer die gleiche Summe 101 zu bekommen. Da dies fünfzigmal passiert, hat man als Gesammtsumme 5050. Gauß bewährte damit zum erstenmal an einem Beispiel sein hervorragendes Geschick in der Ausführung von Zahlenrechnungen. Für ihn war jede Rechnung mit reichen Beobachtungen am Zahlenmaterial und mit mathematischen Einsichten verbunden, und das ermöglichte ihm stets ein müheloses unterhaltendes Finden der Ergebnisse. Man darf den kleinen Fund nicht unterschätzen. Denn Gauß war immerhin erst acht Jahre alt. Universitätslehrer machen immer wieder die Erfahrung, daß selbst junge Studenten der Mathematik nicht immer von allein auf dies Berfahren zur Ermittlung der Summe auseinanderfolgender ganzer Zahlen kommen. Dies Ereignis und wohl noch manche andere Erfahrung mit dem frühreisen Kind ließ in Büttner den Entschluß reisen, dem kleinen Gauß eine besondere Förderung widerfahren zu lassen. Er befreite ihn von seinem normalen Rechenunterricht und ließ ihn mit seinen Gehilfen Bartels zusammenarbeiten, der zufällig gleichfalls mathematsch begabt und mathematisch interessiert war. Dieser damals etiva sechzehnjährige Jüngling studierte nun bald mit Gauß höhere mathematisch Werte und trieb mit ihm auch sprachliche Studien, da er sich selber auf den Besuch des Kollegium Carolinum—heute Technische Hochschule—vorbereitete, in das er 1788 auch eintrat.

Most of us know the story of Gauss who, when his teacher asked the class to sum the integers from 1 to 100, quickly noted that this was 50×101=5050, and was the only student to obtain the correct answer!

Dorothea no doubt noticed the first signs of her son's genius in the events that are recalled as anecdotes from Gauss' early years: how, for instance, when he was three, his discovered a mistake in his father's calculation of the wages for one of his servants. Another event occurred when her son was seven and attended the Katharina-school. The teacher had set the class the task of calculating the sum 1 + 2 + 3 + .... + 100 — probably to get a bit of peace for himself. Carl saw the trick, wrote the answer on his slate and handed it in. When at last all pupils had finished the work, Carl having waited all the time with his arms crossed, his answer proved to be correct, much to the teacher's astonishment.

2- La stupéfaction de monsieur Büttner

Johann Carl Friedrich Gauss vécut de 1777 à 1855 en Allemagne et fut un talentueux physicien (électrostatique), mathématicien, astronome et géodésien. Il avait dès son enfance stupéfié par son génie son maître d'école alors qu'il était âgé de seulement 7 ans. Comment cela s'est-il passé?

Le maître d'école Büttner avait coutume dit-on, de poser à ses élèves le problème qui consiste à calculer la somme des nombres entiers allant de 1 jusqu'à 100 (1 + 2 + 3 + ... + 99 + 100 = combien?). Bien évidemment c'est là un problème laborieux et qui nécessite de la concentration, de la persévérance et beaucoup de temps d'autant plus qu'il s'adresse à de jeunes élèves. Lorsque le jeune Gauss au bout de quelques minutes seulement après le début de l'exercice annonça qu'il trouve 5050, Mr. Büttner resta pantois devant l'exactitude du résultat, trouvé avec une célérité hors du commun. Comment Gauss s'y est-il pris? Il a "tout simplement" fait appel au potentiel fort élevé de la fonction cognitive que recèle la représentation graphique.

3- la représentation mentale et graphique de Gauss

Ce qui nous fait dire aujourd'hui que Gauss avait du génie n'est pas tant le résultat qu'il trouva en un temps qui défia toute imagination, mais c'est plutôt l'élégance de la méthode graphique qu'il utilisa pour trouver son résultat. Mais alors comment a-il fait pour obtenir cette méthode?

Répondre à cette question n'est certainement simple, mais l'on peut avancer que la présentation d'une solution élégante a pour préalable une bonne compréhension du problème. Aujourd'hui, la psychologie cognitive définie la compréhension comme étant une opération mentale impliquant des processus complexes parmi lesquelles la représentation mentale occupe une place de choix.

Proposons nous maintenant d'analyser la solution de Gauss. Il a certainement du commencer par faire comme tous ses camarades: additionner 1 + 2 = 3, pour faire ensuite 3 + 3 = 6, ainsi de suite. Mais il a du se rendre compte assez vite que cette façon de faire était fastidieuse, et allait prendre beaucoup trop de temps et c'est probablement pour cette raison qu'il a renoncé à cette procédure classique.

La philosophie nous enseigne que c'est par l'intuition qu'on découvre et l'intuition de Gauss pour le problème qui nous occupe était de faire appel à un arrangement spatial approprié des nombres qui permet de trouver la solution. En effet, la somme recherchée demeure inchangée si l'on adopte la disposition classique:

1+2+3+ ... +97+98+99+100 = Somme recherchée

         1  +   2  +   3  + ... +  48  +  49  + 50  +
 100 +  99  +  98  +  97  + ... +  52  +  51
 100 + 100  + 100  + 100  + ... + 100  + 100  + 50

Mais l'avantage avec la deuxième disposition c'est que l'addition faite verticalement donne: 100 ajoutés à 1 + 99 = 100, ajoutés encore à 2 + 98 = 100 etc. Nous avons en définitive: 50 x 100 = 5000 et 5000 + 50 = 5050.

Carl Friedrich Gauss (1777–1855), unlike the men discussed in the preceding chapter, was an infant prodigy. His father was an upright but autocratic Brunswick cooper who died shortly before Gauss's thirty-first birthday. His mother outlived her husband by another thirty-one years, and she resided with Carl Friedrich and his family for most of that time. Gauss enjoyed numerical computation as a child; an anecdote told of his early schooling is characteristic: One day, in order to keep the class occupied, the teacher had the students add up all the numbers from one to a hundred, with instructions that each should place his slate on a table as soon as he had completed the task. Almost immediately Carl placed his slate on the table, saying, "There it is." The teacher looked at him scornfully while the others worked diligently. When the instructor finally looked at the results, the slate of Gauss was the only one to have the correct answer, 5050, with no further calculation. The ten-year-old boy evidently had computed mentally the sum of the arithmetic progression 1 + 2 + 3 + ... + 99 + 100, presumably through the formula m(m+1)/2. His teachers soon called Gauss's talent to the attention of the Duke of Brunswick....

An edifying classroom tale about the childhood of a great mathematician can serve as a starting point. To keep the bored and unruly schoolboy Karl Friedrich Gauss busy for a good long time while teaching arithmetic to his mates, his master assigned him the task of adding up all the whole numbers from 1 through 100. The boy paused just a moment and answered 5050, which is, of course, correct. (Of course?) Gauss was not an idiot-savant. How did he do it? He instantly recognized that this regular sequence of 100 numbers could be arranged, starting at each end, into 50 pairs, each of which (1 + 100, 2 + 99, etc.) summed to 101. 50 times 101 equals 5050. (Some accounts tell that it was a different, and superficially more complicated, arithmetic series of 100 numbers, but the principle of the solution remains the same.) The schoolmaster had counted on young Gauss having to add each number in sequence—a long chain of simple calculations that could not be simplified. The boy genius, instead, exploited the highly ordered state of the hundred numbers that he had been given to add up.

XLII. propositio de scala habente gradus centum. Est scala una habens gradus c. In primo gradu sedebat columba una; in secundo duae; in tertio tres; in quarto iiii; in quinto v. Sic in omni gradu usque ad centesimum. Dicat, qui potest, quot columbae in totum fuerunt?

42. proposition concerning the ladder having 100 steps. There is a ladder which has 100 steps. One dove sat on the first step, two doves on the second, three on the third, four on the fourth, five on the fifth, and so on up to the hundredth step. Let him say, he who can, How many doves were there in all?

Solutio. Numerabitur autem sic: a primo gradu in quo una sedet, tolle illam, et junge ad illas xcviiii, quae nonagesimo [nono] gradu consistunt, et erunt c. Sic secundum ad nonagesimum octavum et invenies similiter c. Sic per singulos gradus, unum de superioribus gradibus, et alium de inferioribus, hoc ordine conjunge, et reperies semper in binis gradibus c. Quinquagesimus autem gradus solus et absolutus est, non habens parem; similiter et centesimus solus remanebit. Junge ergo omnes et invenies columbas vl.

Solution. There will be as many as follows: Take the dove sitting on the first step and add to it the 99 doves sitting on the 99th step, thus getting 100. Do the same with the second and 98th steps and you shall likewise get 100. By combining all the steps in this order, that is, one of the higher steps with one of the lower, you shall always get 100. The 50th step, however, is alone and without a match; likewise, the 100th stair is alone. Add them all and you will find 5050 doves.

At the age of 10, Carl Gauss, who would go on to become one of the most prolific mathematical geniuses ever, was simply one of many boys taking a class in arithmetic. His teacher, intending to put everyone to work for a while, told the boys to sum all of the counting numbers from 1 to 100, and they set to work. Within a few seconds, Gauss stood up with his slate, walked to the teacher's desk, laid it down (as was the custom for students who had finished a long problem), and announced, "There it lies!" The teacher was incredulous. Gauss had written the number 5050, and nothing else.

Asked to explain himself, Gauss said he noticed that the sum of the first and last numbers is 101 (1 + 100), and that each pair working in from the outside also summed to 101—that is, 2 + 99, 3 + 98, 4 + 97, all the way to 50 + 51. Seeing that there were 50 such pairs, he multiplied 50 by 101 to get 5050. This reasoning, which can be applied to any arithmetic sequence, leads to a general formula for such a sum: S = n∗(a+b)/2, where a is the first term, b the last, and n the number of terms.

Gauss was one of those remarkable infant prodigies whose natural aptitude for mathematics soon becomes apparent. As a child of three, according to a well-authenticated story, he corrected an error in his father's payroll calculations. His arithmetical powers so overwhelmed his schoolmasters that, by the time Gauss was 10 years old, they admitted that there was nothing more they could teach the boy. It is said that in his first arithmetic class Gauss astonished his teacher by instantly solving what was intended to be a "busy work" problem: Find the sum of all the numbers from 1 to 100. The young Gauss later confessed to having recognized the pattern

   1    2   3  ... n-1  n
   n  n-1 n-2  ...   2  1

G. war ein Kind von wunderbar fruhreifer Entwicklung. Nicht oft mag es vorkommen, daß ein Kind das Lesen von selbst erlernt, indem es die Bedeutung der einzelnen Buchstaben bald diesem, bald jenem Hausgenoffen abtragt. Faft unglaublich erfcheint die gut verbürgte Geschichte, daß das dreijährige Kind zuhörend, wie der Vater Taglöhner für stundenweise Arbeit ablohnte, die Auszahlung mit dem Zuruse unterbrach, die Summe sei nicht richtig, es betrage so viel, und daß seine Angabe bei wiederholt angestellter Rechnung sich als die richtige erwies. Ein kleines Ereigniß von für den Bildungsgang von G. bedeutendster Tragweite war folgendes: Er war eben 9 Jahre alt, als er 1786 in die Rechenschule kam. Die erste Ausgabe, welche Büttner, der wegen seiner Strenge gefürchtete Lehrer, den Schülern vorlegte, betraf die Addition von Zahlen, welche eine arithmetiche Reihe bildeten. Kaum hatte der Knabe den Wortlaut der Ausgabe gehört, so schrieb er zuerst von allen Schülern ohne jegliche Zwischenrechnung die Endumme auf seine Tafelund legte sie, wie es eingeführt war, umgedreht auf den Schultisch in die Mitte des Zimmers. Als alle Tafeln so abgegeben waren und verglichen wurden, war die Zahl des kleinen voreiligen Schreibers eine von den wenigen richtigen. Er entging so nicht blos der ihm für seine Leichtfertigkeit zugedachten gründlichen Bekanntschaft mit der Reitpeitsche des Lehrers, Büttner lieb sogar selbst ein besseres Rechenbuch aus Hamburg kommen, um es dem Knaben zu geben.

A story about the young Karl Friedrich Gauss well illustrates the importance of identifying paradigms for (mental) action. When children in his class were asked to sum the numbers one through ten, most used the procedure 1+2=3, 3+3=6, 6+4=10, ... , 45+10=55. Depending upon their proficiency in addition, some were speedy and some were slow; some were accurate and some were not. Karl Friedrich conceived of the problem instead as one with five 10s (1+9, 2+8, 3+7, 4+6, and 10) and one 5. He was quick to reach the answer not necessarily because of his facility in addition, but because he saw the situation differently than his classmates.

The next example is one that is associated with Carl Friedrich Gauss. As one version of the story goes, when Gauss was 10 years old his teacher, Herr Büttner, asked the students to sum the integers from 1 to 100. Gauss did it almost instantly. It is believed that he did it by the following method.

Write the sum horizontally forwards and backwards as:

  1 +  2 +  3 + ... + 99 + 100
100 + 99 + 98 + ... +  2 +   1

When Gauß was a boy, he visited a small school with several age-groups in one class-room. To keep Gauß's age-group busy while talking to another group, the teacher asked them to add up all of the (natural) numbers from 1 through 100. Almost immediately, Gauß wrote "5050" on his slate and put it on the teacher's desk with the classical words "Ligget se!" ("[there] it lies", in his dialect). Needless to say that he had discovered the n×(n+1)/2 rule, for himself, in this very moment.

Gauss, another exceptional mathematician as well as a calculating prodigy, is credited with a similar performance at a young age. His teacher asked his class to add all the numbers from 1 to 100, probably hoping to keep his pupils quiet for a half-hour. But little Gauss immediately raised his slate with the result. He had rapidly perceived the symmetry of the problem. By "mentally folding" the number line, he could group 100 with 1, 99 with 2, 98 with 3, and so on. Hence the sum was reduced to 50 pairs, each totalling 101, for a grand total of 5,050.

Gauss came from extremely humble origins. His grandfather was a landless peasant; his father was a jobbing gardener and bricklayer. Gauss attended the poorest kind of local school. A famous incident, reported from that school, is much more likely to be true than most such stories are. One day the schoolmaster, to give himself a half-hour break, set the class to adding up the first 100 numbers. Almost instantly, Gauss threw his slate onto the master's table, saying, "Ligget se!" which in the peasant dialect of that place and time meant, "There it is!" Gauss had mentally listed the numbers horizontally in order (1, 2, 3, ..., 100), then in reverse order (100, 99, 98, ...,1) then added the two lists vertically. (101, 101, 101, ... , 101). That is 100 occurrences of 101, and since all the numbers were listed twice, the required answer is half this sum: 50 times 101, which is 5,050. Easy when you have been told it, but not a method that would occur to the average 10-year-old; not even the average 30-year-old, for that matter.

Born in Brunswick, Germany, in 1777, Karl Friedrich Gauss displayed immense mathematical talent from a very early age. Stories tell of him being able to maintain his father's business accounts at age three. According to another story, while in the elementary school, Gauss confounded his teacher by observing a pattern that enabled him to avoid a decidedly tedious calculation.

Gauss' teacher had asked the class to add together all the numbers from 1 to 100. Presumably the teacher's aim was to keep the students occupied for a time while he was engaged in something else. Unfortunately for him, Gauss quickly spotted the following shortcut to the solution.

You write down the sum twice, once in ascending order, then in descending order, like this:

  1 +   2 +   3 + ... +  98 +  99 + 100
100 +  99 +  98 + ... +   3 +   2 +   1.

101 + 101 + 101 + ... + 101 + 101 + 101.

100 × 101 = 10,100.

Gauss' trick works for any number n, not just 100. In the general case, when you write the sum from 1 to n in both ascending and descending order and add the two sums column by column, you end up with n copies of the number n+1, which is a total of n(n+1). Halving this total gives the answer:

1 + 2 + 3 + ... + n = n(n+1)/2.

It is interesting to note that the formula on the right-hand side of the above identity also captures a geometric pattern. Numbers of the form n(n+1)/2 are called triangular numbers, since they are exactly the numbers you can obtain by arranging balls in an equilateral triangle.

Born in 1777 in Brunswick, Carl Friedrich Gauss showed early and unmistakable signs of being an extradordinary youth. As a child of three, he was checking, and occasionally correcting, the books of his father's business, this from a lad who could barely peer over the desktop into the ledger. A famous and charming story is told of Gauss's elementary school training. One of his teachers, apparently eager for a respite from the day's lessons, asked the class to work quietly at their desks and add up the first hundred whole numbers. Surely this would occupy the little tykes for a good long time. Yet the teacher had barely spoken, and the other children had hardly proceeded past "1 + 2 + 3 + 4 + 5 = 15" when Carl walked up and placed the answer on the teacher's desk. One imagines that the teacher registered a combination of incredulity and frustration at this unexpected turn of events, but a quick look at Gauss's answer showed it to be perfectly correct. How did he do it?

First of all, it was not magic, nor was it the ability to add a hundred numbers with lightning speed. Rather, even at this young age, Gauss exhibited the penetrating insight that would remain with him for a lifetime. As the story goes, he simply imagined the sum he sought—which we shall denote by S—being written simultaneously in ascending and in descending order:

S =   1 +  2 +  3 +  4 + ... + 98 + 99 + 100
S = 100 + 99 + 98 + 97 + ... +  3 +  2 +   1.

2S = 101 + 101 + 101 + ... +  101 + 101 + 101

S = 1 + 2 + 3 + 4 + ... + 99 + 100 = 10100/2 = 5050

All this went through Gauss's little head in a flash. It was clear that he was going to make a name for himself.

Even in his earliest years Gauss gave extraordinary proofs of his mental ability. After he had asked various members of the household about the pronunciation of letters of the alphabet, he learned to read by himself, we are told, even before he went to school, and showed such remarkable comprehension of number relationships and such an incredible facility and correctness in mental arithmetic that he soon attracted the attention of his parents and the interest of intimate friends.

Gauss' father carried on in the summer what we would call today a bricklayer's trade. On Saturday he was accustomed to give out the payroll for the men working under him. Whenever a man worked overtime he was, of course, paid proportionately more. Once, after the "boss" had finished his calculations for each man and was about to give out the money, the three-year-old boy got up and cried in childish voice: "Father, the calculation is wrong," and he named a certain number as the true result. He had been following his father's actions unnoticed, but the figuring was carefully repeated and to the astonishment of all present was found to be exactly as the little boy had said. Later Gauss used to joke and say that he could figure before he could talk.

Gauss entered the St. Katharine's Volksschule in 1784, after he had reached his seventh year. Here elementary instruction was offered, and the school was under the direction of a man named J. G. Büttner. The schoolroom was musty and low, with an uneven floor. From the room one could look on one side toward the two tall, narrow Gothic spires of St. Katharine's Church, on the other toward stables and the rear of slums. Here Büttner, the whip in his hand, would go back and forth among about two hundred pupils. The whip was recognized by great and small of the day as the ultima ratio of educational method, and Büttner felt himself justified in making unsparing use of it according to caprice and need. In this school, which seems to have had the cut and style of the middle ages, young Gauss remained for two years without any incident worth recording.

Eventually he entered the arithmetic class, in which most pupils remained until their confirmation, that is, until about their fifteenth year. Here an event occurred which is worthy of notice because it was of some influence on Gauss' later life, and he often told it in old age with great joy and animation.

Büttner once gave the class the exercise of writing down all the numbers from 1 to 100 and adding them. The pupil who finished an exercise first always laid his tablet in the middle of a big table; the second laid his on top of this, and so forth. The problem had scarcely been given when Gauss threw his tablet on the table and said in Brunswick low dialect: "Ligget se" (There 'tis). While the other pupils were figuring, multiplying, and adding, Büttner went back and forth, conscious of his dignity; he cast a sarcastic glance at his quick pupil and showed a little scorn. In the end, however, he found on Gauss' tablet only one number, the answer, and it was correct. But the young boy was in a position to explain to the teacher how he arrived at this result. He said: "100+1=101; 99+2=101, 98+3=101, etc., and so we have as many 'pairs' as there are in 100. Thus the answer is 50×101, or 5,050." Gauss sat quietly, firmly convinced that his problem had been correctly solved, just as he later did in the case of any completed piece of work. Many of the other answers were wrong and were at once "rectified" by the whip.

The first sequence of numbers above [1, 3, 6, 10, 15,...] consist of what are called the triangular numbers. The tenth number on the list is the number of beans required to build a triangle with ten rows, starting with one bean in the first row and ending with ten beans in the last row. So the Nth triangular number is got by simply adding the first N numbers: 1 + 2 + 3 + ... + N. If you want to find the 100th triangular number, there is a long laborious method in which you attack the problem head on and add up the first 100 numbers.

Indeed, Gauss's schoolteacher liked to set this problem for his class, knowing that it always took his students so long that he could take 40 winks. As each student finished the task they were expected to come and place their slate tablets with their answer written on it in a pile in front of the teacher. While the other students began laboring away, within seconds the ten-year-old Gauss had laid his tablet on the table. Furious, the teacher thought that the young Gauss was being cheeky. But when he looked at Gauss's slate, there was the answer — 5,050 — with no steps in the calculation. The teacher thought that Gauss must have cheated somehow, but the pupil explained that all you needed to do was put N=100 into the formula 1/2 × (N + 1) × N and you will get the 100th number in the list without having to calculate any other numbers on the list on the way.

Rather than tackling the problem head on, Gauss had thought laterally. He argued that the best way to discover how many beans there were in a triangle with 100 rows was to take a second similar triangle of beans which could be placed upside down on top of the first triangle. Now Gauss had a rectangle with 101 rows each containing 100 beans. Calculating the total number of beans in this rectangle built from the two triangles was easy: there are in total 101 × 100 = 10,100 beans. So one triangle must contain half this number, namely 1/2 × 101 × 100 = 5,050. There is nothing special here about 100. Replace it by N and you get the formula 1/2 × (N + 1) × N.

Așa a fost și la școala elementară din Braunschweig. Carl Friedrich era pasionat după orele de socotit. E drept, pe atunci în învățământ se folosea mai mult învățatul pe dinafară și nu se dădeau explicații cu privire la raporturile mai profunde.

Profesorul Buttner, jucându-se cu nelipsita nuia, se opri în intervalul dintre șirurile de bănci.

- Și acuma, copii, aveți următoarea temă: să adunați toate numerele de la 1 la 40. Cine termină, îmi aduce tăblița la catedră. În clasă se făcu liniște și capetele se aplecară spre bănci.

''O să le trebuiască ceva timp pentru asta, se gândi profesorul. Numai bine ca să mă mai odihnesc și eu nițel.''

Se duse la catedră, dar abia pusese jos nuiaua și se așezase pe scaun că micul Carl Friedrich sări din bancă și veni spre el.

- Am terminat! strigă el fericit și așeză tăblița cu partea scrisă în jos, așa cum era obiceiul, în fața profesorului înmărmurit de uimire.

''Ei, își zise el, cine știe ce prostie a mai făcut și băiatul asta în graba lui''. Și aruncă o privire batjocoritoare spre Gauss, care aștepta victorios în banca lui.

A durat destul de mult până când au calculat toți elevii, adunând cu mare greutate cele patruzeci de numere. Tăblițele se strângeau încet pe catedra profesorului. Acesta le întorcea una după alta cu un zâmbet ironic. Dar zâmbetul se transformă în uluire când ajunse la prima tăbliță și citi pe ea rezultatul corect: 820!

Aici nu erau adunate numerele unul după altul la nesfârșit, ca pe celelalte tăblițe. Privirea experimentată descoperise imediat legăturile dintre numere. În felul cum le aranjase Gauss, numerele păreau că dansează.

Nemaipomenit!

Băiatul adunase un număr de la început cu unul de la sfârșit: 1+40, 2+39, 3+38, 4+37 până când a format douăzeci de perechi a căror suma făcea mereu 41. Nu mai rămânea acum decât să înmulțească 41 cu 20 și rezultatul corect era obținut.

Profesorul nu-și reveni multă vreme din uimire. Ba chiar își reproșa în sinea lui atitudinea de mai înainte. Căci băiatul care se foia nerăbdător în banca lui, descoperise cu propriile lui forțe metoda adunării numerelor dintr-un șir natural, fără să-i fi spus nimeni nimic despre asta vreodată.

Mai târziu, când s-a ivit o ocazie, profesorul i-a adus de la Hamburg aritmetica Remers Arithmetica. I-o dărui băiatului pentru ca să-și poată domoli setea de cunoștințe mai bine decât în timpul orelor lui de aritmetică.

Fericite erau clipele pe care talentatul Carl Friedrich le petrecea cu manualul de la Hamburg. ''O carte dragă'' - scrisese mânuța copilului pe coperta interioară a cărții.

We motivate the need for induction using a story about the mathematician Gauss when he was 10. His old-fashioned arithmetic teacher liked to show off to his students by asking them to add a large number of sequential numbers by hand, which the teacher knew from a book could be done quickly using the formula

1 + 2  +3 +...+ (n-1) + n = (n(n+1))/2.      (3.1)

By the way, long sums of numbers arise in integration and in models such as computing compound interest on a savings account or adding up populations of animals. Addition formulas like (3.1) are therefor practically useful, which is why we are interested in them.

The teacher posed the sum 1 + 2 + ... + 99 to the class, and almost immediately Gauss came up and laid his slate down in the desk with the correct answer, 4950, while the rest of the class still struggled. How did young Gauss manage to compute the sum so quickly? He did not know the formula (3.1), he just derived it using the following clever argument. To sum 1 + 2 + ... + 99, we group the numbers two by two as follows:

1 + ... + 99
= (1 + 99) + (2 + 98) + (3 + 97) + ... (49 + 51) + 50
= 49 × 100 + 50 = 49 × 2 × 50 + 50 = 99 × 50

Gauss very early in life exhibited a remarkable cleverness with numbers, becoming a "wonder child" at the age of two. There are a couple of oft-told stories illustrating the boy's unusual ability.

One of the stories tells how on a Saturday evening Gauss's father was making out the weekly payroll for the laborers of the small bricklaying business that he operated in the summer. The father was quite unaware that his young three-year-old son Carl was following the calculations with critical attention, as so was surprised at the end of the computation to hear the little boy announce that the reckoning was wrong and that it should be so and so instead. A check of the figures showed that the boy was correct, and on subsequent Saturday evenings the youngster was propped up on a high stool so that he could assist with the accounts. Gauss enjoyed telling this story later in life, and used to joke that he could figure before he could talk.

The other story dates from Gauss's schooldays, when he was about ten years old. At the first meeting of the arithmetic class, Master Büttner asked the pupils to write down the numbers from 1 through 100 and add them. It was the custom that the pupils lay their slates, with their answers thereon, on the master's desk upon completion of the problem. Master Büttner had scarcely finished stating the exercise when young Gauss flung his slate on the desk. The other pupils toiled on for the rest of the hour while Carl sat with folded hands under the scornful and sarcastic gaze of the master. At the conclusion of the period, Master Büttner looked over the slates and discovered that Carl alone had the correct answer, and upon inquiry Carl was able to explain how he had arrived at his result. He said, "100+1=101, 99+2=101, 98+3=101, etc., and so we have as many 'pairs' as there are in 100. Thus the answer is 50 × 101, or 5050."

At school, Gauss showed little of his precocious talent until the age of nine, when he was admitted to the arithmetic class. The master had set what appeared to be a complicated problem involving the addition of a series of numbers in arithmetical progression. Although he had never been taught the simple formula for solving such problems, Gauss handed in his slate within seconds. For the next hour the boy sat idly while his classmates labored. At the end of the lesson there was a pile of slates on top of Gauss's, all with incorrect answers. The master was stunned to find at the bottom the slate from the youngest member of the class bearing the single correct number. He was so impressed that he bought the best available arithmetic textbook for Gauss and thereafter did what he could to advance his progress.

Gauss was born in Brunswick, Germany as the only son of poor peasants living in miserable conditions. He exhibited such early genius that his family and neighbors called him the "wonder child". When he was two years old, he gradually got his parents to tell him how to pronounce all the letters of the alphabet. Then, by sounding out combinations of letters, he learned (on his own) to read aloud. He also picked up the meanings of the number symbols and learned to do arithmetical calculations. The story as told by Eric T. Bell:

"One Saturday Gerhardt Gauss was making out the weekly payroll for the laborers under his charge, unaware that his young son was following the proceedings with critical attention. Coming to the end of his long computations, Gerhardt was startled to hear the little boy pipe up, 'Father, the reckoning is wrong, it should be ..." A check of the account showed that the figure named by the young Gauss was correct." [Eric Temple Bell, Men Of Mathematics , Simon Schuster, Inc., New York, 1937]

When Gauss was ten years old he was allowed to attend an arithmetic class taught by a man (Buttner) who had a reputation for being cynical and having little respect for the peasant children he was teaching. The teacher had given the class a difficult summation problem in order to keep them busy and so that they might appreciate the "shortcut" formula he was preparing to teach them. Gauss took one look at the problem, invented the shortcut formula on the spot, and immediately wrote down the correct answer. This act was apparently so astonishing that Herr Buttner was transformed into a champion for this young boy. "Out of his own pocket he paid for the best textbook on arithmetic obtainable and presented it to Gauss. The boy flashed through the book." (E. T. Bell). Buttner, realizing that he could teach this young genius no more, recommend him to the Duke of Brunswick, who granted him financial assistance to continue his education into secondary school and finally into the University of Gottingen.

Even as a child, Gauss showed a remarkable skill with numbers—in particular the set of natural numbers {1, 2, 3, 4, ...}. According to mathematical lore, one day his teacher asked the class to add all the natural numbers from 1 to 100. Students were instructed to place their slates on the table when finished. To the surprise of the teacher, young Gauss placed his slate on the table after only a few moments.

To find the sum of the first 100 natural numbers, Gauss used a method involving a finite series. For example, the sum of the first 100 numbers can be written as the arithmetic series S100:

S100 = 1 + 2 + ... + 99 + 100

S100 = 100 + 99 + ... + 2 + 1

 S100 =     1  +    2  +  ...  +   99  +  100
 S100 =   100  +   99  +  ...  +    2  +    1
2S100 =   101  +  101  +  ...  +  101  +  101

2S100 = 100(101)

S100 = (100(101))/2 = 5050

Therefore, let us consider a numerical problem, which has its roots in the following story about the young Carl Frederick Gauss, who was to become one of the greatest mathematicians of all time. The story is probably apocryphal, but is still a good story. This version follows Polya (1962), who also uses it to introduce recursion in mathematical problems.

When Gauss was in primary school, the teacher, hoping to keep his students occupied while he attended to other matters, gave a tough problem: to add the numbers 1, 2, 3, up to 20. While the other children were just getting started, young Gauss walked to the teacher's desk and handed in his slate. The teacher, thinking this an act of impudance, did not even bother to look at Gauss's work until all the other children had handed in theirs. When he did look at it, he was surprised to find that it contained just a single number, the right answer.

I will not consider how Gauss did this, but I commend Polya's delightful book to you.

Auf die Entwicklung dieser außerordentlichen Gabe und zugleich auf die Betätigung seines Geistes dabei wirft eine andre Geschichte ein helleres Licht. Den Schülern der under des Lehrers Büttner Leitung stehenden Rechenklasse der Katharineenschule in Braunschweig wurde die Aufgabe vorgelegt, die Summe einer Reihe auf einander folgende Zahlen zu bilden. Jeder, der die Rechnung beendet haben würde, sollte die Tafel auf einen Sammeltisch legen. Kaum war die Aufgabe gestellt, so legte der damals neunjährige Gauß seine Tafel mit den Worten: Dar licht se! hin. Der alte Büttner musterte den schnellfertigen Knaben mit spöttischem Mitleid, während die andern Schüler die Stunde hindurch weiter Rechneten. Auf der Tafel von Gauß stand nur eine Zahl, das richtige Ergebnis. Er hatte das Summationsprinzip für die arithmetischen Reihen auf den ersten Blick herausgefunden.

Am 30. April 1777 in Braunschweig als Sohn eines Gassenschlächters geboren, verblüffte Carl Friedrich Gauss — der von sich selbst sagte, er habe eher rechnen als sprechen gelernt — schon als Kind seine Lehrer. In der mit 100 Schülern überfüllten Schulstube erteilte der Lehrer die Aufgabe, alle Zahlen von 1 bis 100 zu addieren. Lange vor seinen Mitschülern hatte der kleine Carl Friedrich das richtige Ergebnis parat. Anhand von 50 Zahlenpaaren mit der Summe von 101 (1+100, 2+99, 3+98 und so weiter) löste er die Aufgabe mit 50 x 101 = 5050 als richtiges Ergebnis.

Let's start with a simple example: Gauss's Formula for the sum of integers.

Johann Carl Friedrich Gauss was born on April 30, 1777, in Brunswick, Germany. At the age of seven, Carl Friedrich Gauss started elementary school, and his potential was noticed almost immediately. His teacher, Büttner, and his assistant, Martin Bartels, were amazed when Gauss summed the integers from 1 to 100 instantly by spotting that the sum was 50 pairs of numbers each pair summing to 101. Gauss's Formula for the sum of integers was born.

<?php
    $result = gauss(4);
    echo "Sum from 1 to 4: $result<br>\n";

# function for calculating the sum from 1 to $upper
function gauss($upper)
{
    if    (is_int($upper) && ($upper > 0))
      {
       return($upper*($upper+1)/2);
      }
}
?>

Gauss was born on April 30th, 1777 in the Duchy of Brunswick, now a part of Germany. He was a child prodigy, and many stories are told of his early mathematical prowess. It is well-documented that he corrected an error in his father's payroll calculations at the age of three, and as an adult he explained that by his recollection he could count before he could talk. Probably the most famous story about young Gauss occured in 1786, when he was nine years old. His teacher, J. G. Büttner, assigned his class the task of adding all of the numbers from 1 to 100. Gauss turned in his slate after only a few seconds, with only the final answer written down. Büttner studiously ignored him until the class had finished, and was astonished to find that Gauss's answer was correct. He asked Gauss how he had arrived at his answer, and Gauss explained: "100+1=101; 99+2=101; 98+3=101, etc., and so we have as many pairs as there are in 100. Thus the answer is 50 × 101 = 5050." Büttner realized at this point that he was not dealing with a typical student, and took steps to assure Gauss's education.

After Gauss' death in February of 1855, a medal was struck in his honor... with the inscription Mathematicorum princeps (Prince of Mathematicians) under his bas-relief. The history of every real prince begins with a childhood surrounded by legends. Gauss was not an exception....

At the age of seven, Carl Friedrich entered Catherine's School. In that school students were not taught how to count until the third grade, so for the first two years nobody paid attention to little Carl.

The children usually got to the third grade at the age of 10 and stayed in that grade until confirmation (at the age of 15). The teacher Büttner had to devote himself simultaneously to children of different ages and knowledge. For this reason, he often gave some of the students long exercises in calculation in order to be able to talk to other students. Once, he asked a group of students, among them Gauss, to sum up all natural numbers from 1 to 100. As a student finished the calculations, he would place his slate on the teacher's desk. The order of the slates was taken into account when giving marks. Ten-year-old Gauss turned in his slate as soon as Büttner had finished assigning the task. To everybody's surprise, only Gauss' answer turned out to be correct. The explanation was simple: as the teacher had been dictating the task, Gauss found a trick for summing a general arithmetic progression! The fame of the infant prodigy spread all over Brunswick.

Most of us imagine mathematicians to be old people with beards and thick glasses, yet many of the important mathematical discoveries have been made by fairly young people.

One of the youngest and most famous mathematicians in all of history was Carl Friedrich Gauss who was born in Germany in 1777 and died in 1855.

He came from a poor family. His father was a gardener and his mother a housekeeper. Young Gauss showed his mathematical ability at a very early age. When he was three years old he watched his father add up a long column of numbers. Gauss pointed out an error and gave his father the correct answer. When the father checked the addition, he found his son was indeed correct.

When Gauss was ten years old he began his first lessons in arithmetic. The teacher gave the class a long and difficult problem so they would have to spend hours to find the answer.

The problem involved adding up a sequence of numbers like: 1 + 2 + 3 + 4 + 5 + 6 + ... + 999 + 1000 = ? There is a trick to solve problems like this but it was unknown to the young students.

However, Gauss discovered the trick for himself and quickly solved the problem while all the other students worked for hours and all the answers were wrong except for Gauss's. Recognizing that Gauss was special, his teacher helped him to advance in his studies.

... Most mathematicians are no less bored by adding up long columns of figures than the rest of us. They do not consider it their job, and are usually annoyed when nonmathematicians assume that it is.

The point may be illustrated by two episodes in the life of Karl Gauss (1777–1855), one of the greatest of mathematicians. Gauss was born a poor boy, the son of a bricklayer, in Braunschweig, Germany. The schoolmaster in the local school Gauss attended, a certain Herr Büttner, was a hard taskmaster who gave his classes practice in arithmetic by asking them to add up long sequences of large numbers. For his own convenience, so that he would not have to do the tedious arithmetic involved to check his pupils' almost invariably erroneous answers, the sequences of numbers he assigned his classes to add were chosen to form what it called an arithmetic series—the successive numbers in the long list differed by a constant amount. For example, the series 11, 14, 17, 20, 23, 26 is such a series, in which each term increases by 3. Büttner then makes use of a well-known formula for the sum of such a series: the sum is equal to the number of terms times one-half the sum of the first and last terms. For the series given above, the sum is

6 * (11 + 26)/2 = 111

In any event, Büttner wrote on the blackboard a list of large numbers forming such a series, and after finishing turned around to face the class, expecting as usual to have a free hour or so while his pupils sweated and struggled, to find little Gauss handing in his slate with the correct sum written out. Gauss had recognized the numbers as forming an arithmetic series, figured out on the spot the formula for the sum, and calculated it. Büttner, to his everlasting credit, though no mathematician himself, knew one when he saw one. With his own money he bought Gauss the best textbook on arithmetic then available and brought the boy's abilities to the attention of people who could help him in his career.

somme = (n(n+1))/2

Karl Friedrich Gauss (1771–1855) was one of the finest mathematicians of all time. The son of a bricklayer, it is said that he spotted formulae for certain arithmetic sums for himself at the age of 10. His teacher had a habit of setting the class long strings of numbers to add up to keep them occupied, all the time knowing a formula for the answer. Gauss outwitted him and all his teacher could do was to buy him a text book and announce that the boy was beyond him.

To evaluate S_n [the sum of the first n positive integers] we can use a trick that Gauss reportedly came up with in 1786, when he was nine years old:

  Sn =   1   +   2   +   3   + ... + (n-1) +   n
 +Sn =   n   + (n-1) + (n-2) + ... +   2   +   1  
 2Sn = (n+1) + (n+1) + (n+1) + ... + (n+1) + (n+1)

Gauss's trick in chapter 1 can be viewed as an application of these three basic laws [i.e., distributive, associative, commutative]. Suppose we want to compute the general sum of an arithmetic progression,

  S = Σ(a+bk).

  S = Σ(a+b(n-k)) = Σ(a+bn-bk).

  2S = Σ((a+bk)+(a+bn-bk)) = Σ(2a+bn).

  2S = (2a+bn)Σ 1 = (2a+bn)(n+1).

  Σ(a+bk) = (a + 1/2bn)(n+1).

Vous avez sûrement déjà entendu parler de Gauss, Carl Friedrich de son prénom, mathématicien allemand... Voici une petite histoire le concernant que vous avez peut-être aussi déjà entendue (elle est célèbre)... mais elle est si belle!

A l'école primaire, Gauss, enfant prodige, agaçait pour le moins son instituteur. Ce dernier pour se "débarasser" de lui, demanda à Gauss de calculer de tête la somme des 50 premiers entiers positifs, c'est-à-dire 1+2+3+4+...+50. L'instituteur pensa ainsi occuper Gauss pour toute la journée. Hélas l'instituteur s'est réjoui trop vite, 5 minutes plus tard Gauss interpella l'instituteur: "1275" fit-il, ce qui laissa l'instituteur bouche-bée.

Mais comment a-t-il fait?

Et bien Gauss remarqua que la somme des "termes symétriques" (de cette somme) est toujours égale à 51: 1+50 = 51 ; 2+49 = 51 ; 3+48 = 51; ... ; 25+26 = 51 et il y a ainsi 25 termes égaux à 51. D'où: 1+2+3+4+...+50 = (1+50)+(2+49)+(3+48)+(4+47)+...+(25+26) = 51×25 = 1275.

Prodigieux...plutôt que de faire l'addition bête et méchante, Gauss avec cette idée de réarrangements des termes ramène le problème à du dénombrement et à une seule opération, une multiplication!

Compris?...petit exercice maintenant: calculer la somme des 999 premiers entiers positifs...de tête bien sûr!

Biographies about or by great men generally contain more or less noteworthy anecdotes, intended to illustrate the budding genius. It is a field in which memory gladly accommodates itself to a fixed path and where imagination easily overtakes the uncertain facts. The situation is especially pernicious in the case of child prodigies, who are often encountered in mathematics, music, and chess. Myths appear with treacherous ease.

Gauss was a mathematical prodigy—it is certain that he was one of the most outstanding examples of this genre, but basically this is unimportant. First-hand accounts of this come from Gauss himself, who in his old age liked to talk of his childhood. From a critical viewpoint they are naturally suspect, but his stories have been confirmed by other persons, and in any case they have anecdotal interest.

During the summers Gebhard Gauss was foreman for a masonry firm, and on Saturdays he used to pay the week's wages to his workers. One time, just as Gebhard was about to pay a sum, Carl Friedrich rose up and said, "Papa, you have made a mistake," and then he named another figure. The three-year-old child had followed the calculation from the floor, and to the open-mouthed surprise of those standing around, a check showed that Carl Friedrich was correct.

Gauss used to say laughingly that he could reckon before he could talk. He asked the adults how to pronounce the letters of the alphabet and learned to read by himself.

When Carl Friedrich was seven years old he enrolled in St. Catherine elementary school. His teacher was J. G. Büttner. The large classroom had a low ceiling, and the schoolmaster walked about on the uneven floor, cane in hand, among his approximately 100 pupils. Caning was the foremost pedagogical aid both for learning and discipline, and Büttner is thought to have used it constantly, either as a consequence of necessity or because of his temperament. Gauss stayed in these surroundings for two years without any ill consequences. It is the traditional picture of that period's public education, when the caning pedagogy was generally accepted—by the adults of course—but we shall soon see that Büttner was more likely above than below average among his colleagues.

When Gauss was about ten years old and was attending the arithmetic class, Büttner asked the following twister of his pupils: "Write down all the whole numbers from 1 to 100 and add up their sum." When the class had a task of that sort they would do the following: the first to finish would go forward to the schoolmaster's desk with his slate and put it down, the next who finished would place his slate upon the first, and so on in a growing pile. The problem is not difficult for a person familiar with arithmetic progressions, but the boys were still at the beginner's level, and Büttner certainly thought that he would be able to take it easy for a good while. But he thought wrong. In a few seconds Gauss laid his slate on the table, and at the same time he said in his Braunschweig dialect: "Ligget se" (there it lies). While the other pupils added until their brows began to sweat, Gauss sat calm and still, undisturbed by Büttner's scornful or suspicious glances.

At the end of the period the results were examined. Most of them were wrong and were corrected with the rattan cane. On Gauss's slate, which lay on the bottom, there was only one number: 5050. (It seems unnecessary to point out that this was correct.) Now Gauss had to explain to the amazed Büttner how he had found his result: 1+100=101, 2+99=101, 3+98=101, and so on, until finally 49+52=101 and 50+51=101. This is a total of 50 pairs of numbers, each of which adds up to 101. Therefore, the whole sum is 50×101=5050. Thus Gauss had found the symmetry property of arithmetic progressions by pairing together the terms as one does when deriving the summation formula for an arbitrary arithmetic progression—a formula which Gauss probably discovered on his own. What this actually entails is that one writes the series both "forward" and "backward"; that is

  1 +  2 + ... + 99 + 100
100 + 99 + ... +  2 +   1

The event is symbolic. For the rest of his life Gauss was to present his results in the same calm, matter-of-fact way, fully conscious of their correctness. The evidence of his struggles would be wiped away from the completed work in the same way. And, like Büttner, many learned persons would wish to be given a detailed explanation, but here a difference would appear, for Gauss would not feel compelled to give one.

Der Fürst der Mathematiker konnte früher rechnen als sprechen... zumindest behauptete er das selbst scherzhaft von sich. Den Anekdoten nach war der am 30. April 1777 in Braunschweig geborene Gauß tatsächlich ein mathematisches Wunderkind, der als dreijähriger bereits den Vater bei der Lohnabrechnung korrigiert haben soll. In der Grundschule berechnete er die Summe der Zahlen von 1 bis 100 nach dem Gesetz s = n(n+1)/2 und als 18jähriger entdeckte er die Konstruktion des regulären Siebzehnecks (mit Zirkel und Lineal).

Auf dieser Seite also glauben die verborgenen Quelladern des Genius riefeln zu hören. Und doch, wie hoch man die Gunst dieser Einflüsse auch anschlagen mag, ein Wunder bleibt es, mit welcher Macht er in diesem Erdenfinde hervorbrach. Ganz ungewöhnlich früh, schon in den Jahren da bei Underen die Seelenvermögen noch im Dunkel der Unbewußtheit schlumern. Aus sich selbst, mit gelegentlicher Nachfrage bei seiner Umgebung, lernt er lesen; am erstaunlichsten aber zeigt sich von frühester Kindheit an die intuitive Kraft seiner Auffassung von Zahlenverhältnissen: er durste scherzend wohl von sich sagen, daß er eher habe rechnen als sprechen sönnen. In seinem dunkeln Heimchenwinkel behorcht der kaum dreijährige Knabe die Berechnungen die der Vater beim Wochenabschluß mit seinen Gefellen anstellt; es handelt sich um die Vergütung von feierabendarbeit nach Verhaltniß des Tagelohnes. Als es ans Unszahlen geht, zirpt er warnend dazwischen, und siehe da, der Alte hat sich verrechnet und was der Kleine angiebt ist das Richtige. In seinem neunten Jahre ist es, daß in der Rechenklasse der Büttnerschen Schule bei St. Katharine, der er seit 1784 angehört, eine arithmetische Reihe summirt werden soll. Die Ausgabe ist kaum gestellt, als Gauß seine Tafel mit einem übermüthigen 'Dar licht se!' auf den Sammeltisch wirst, während alle Underen die Stunde durch rechnen und rechnen. Der alte Büttner mustert den schnellfertigen kleinsten seiner Unglückswürmer mit spöttischem Mitleid: der Bakel wird zu thun bekommen; am Ende jedoch findet er auf Gauß' Tafel nur eine Zahl, das Ergebniß, und es ist richtig. Solche Leistung aeschüttert denn selbst den altern Herrn, den sonst seiner Schaar mit dem ganzen Meisterbewußtsein eines Ludimagister ältern Stiles gagenübersteht; er thut ein Uebriges und verschreibt erpreß für das Wunderkind ein neues Rechenbuch ans Hamburg. Bald genug aber muß er sich zu der Einsicht bequemen, daß es für solchen Schüler bei ihm nichts mehr zu lernen giebt.

Carl Friedrich Gauß wurde am 30. April 1777 in einem kleinen ärmlichen Haus als einziges Kind von Dorothea und Gerhard Diederich Gauß in Braunschweig geboren und zeigte schon sehr früh eine außerordentliche Begabung im Erfassen von Zahlen. Er sagte einmal von sich selber, daß er eher rechnen als lesen konnte. Es zeigte sich aber auch schon früh, daß dies nicht nur eine Begabung für rechnerische Tüfteleien war, sondern daß sich dahinter ein tiefes Verständnis einer Gesetzmäßigkeit in der Welt der Zahlen verbarg bzw. das Vermögen, die Welt der Zahlen als eine geometrische Konstruktion im Geist zu erfassen.

In der Schule hatte der Lehrer die Aufgabe gestellt, alle Zahlen von 1 bis 100 zusammenzuzählen. Er hatte die Aufgabe kaum zu Ende gestellt, schon schreibt der kleine Gauß eine Zahl auf seine Tafel, bringt diese nach vorn und legt sie vor den Lehrer mit den Worten "Ligget se" (Da liegt sie). Die anderen Kinder rechnen die ganze Stunde hindurch und der Lehrer überlegt sich schon, die Strafe für eine solche Frechheit mit dem Rohrstock zu zahlen, doch der kleine Gauß sitzt mit einer ruhigen Sicherheit an seinem Platz und wartet auf das Ende der Stunde. Auf seiner Tafel steht die richtige Zahl 5050, und viele andere sind falsch oder noch nicht fertig.

Er hatte den geometrischen Aufbau der Zahlen sofort vor Augen gehabt und erkannt: Man muß nur die ersten und die letzten Zahlen jeweils verknüpfen, 1+100, 2+99, 3+98... bis 50+51, dann hat man 50mal 101 zusammengezählt und das ist das Ergebnis.

When Gauss—mathematician Karl Friedrich Gauss (1777–1855)—was a 10-year-old boy, his schoolmaster, Buttner, gave the class an arithmetic progression of numbers to add up to keep them busy. We should recall that an arithmetic progression is a sequence of numbers where each number differs from its successor by the same constant. Gauss wrote down the answer just after Buttner finished writing the problem. Although the formula was known to Buttner, no child of 10 had ever discovered it.

For example, suppose we want to add up the seven numbers in the following arithmetic progression:

3, 7, 11, 15, 19, 23, 27

   3    7   11   15   19   23   27
  27   23   19   15   11    7    3
  30   30   30   30   30   30   30

The example illustrates a use of the following formula for the sum of an arithmetic progression of n numbers a₁, a₂, ...., a_n

a1 + a2, + .... + an = n(a1 + an)/2

Insights and connections—that's what mathematicians look for. Carl Friedrich Gauss, who was born in 1777 in Braunschweig, Germany, the son of a masonry foreman, was a master of exposing unsuspected connections. Like Erdös, Gauss was a mathematical prodigy, and in his old age he liked to tell stories of his childhood triumphs. Like the time, at the age of three, he spotted an error in his father's ledger and stopped him just as he was about to overpay his laborers. Like the fact that he could calculate before he could read.

And he certainly could calculate. At the age of ten, he was a show-off in arithmetic class at St. Catherine elementary school, "a squalid relic of the Middle Ages... run by a virile brute, one Büttner, whose idea of teaching the hundred or so boys in his charge was to thrash them into such a state of terrified stupidity that they forgot their own names." One day, as Büttner paced the room, rattan cane in hand, he asked the boys to find the sum of all the whole numbers from 1 to 100. The student who solved the problem first was supposed to go and lay his slate on Büttner's desk; the next to solve it would lay his slate on top of the first slate, and so on. Büttner thought the problem would preoccupy the class, but after a few seconds Gauss rushed up, tossed his slate on the desk, and returned to his seat. Büttner eyed him scornfully, as Gauss sat there quietly for the next hour while his classmates completed their calculations. As Büttner turned over the slates, he saw one wrong answer after another, and his cane grew warm from constant use. Finally he came to Gauss's slate, on which was written a single number, 5,050, with no supporting arithmetic. Astonished, Büttner asked Gauss how he did it, "and when Gauss explained it to him," said Erdös, "the teacher realized that this was the most important event in his life and from then on worked with Gauss always," plying him with textbooks, for which "Gauss was grateful all his life."

What was Gauss's trick? In his mind he apparently pictured writing the summation sequence twice, forward and backward, one sequence above the other:

  1 +  2 +  3  +  4 + ... + 97 + 98 + 99 + 100
100 + 99 + 98  + 97 + ... +  4 +  3 +  2 +   1

"What makes Gauss's method of calculation so special," said Graham, "is that it doesn't just work for this specific problem but can be generalized to find the sum of the first 50 integers or the first 1,000 integers or the first 10,000 integers or whatever number you want. Gauss found a very nice way of showing that if you add all the numbers from one up through any number n, the answer is n times n plus one, all divided by two. This method of summing such a series is really straight from the Book."

Gauss' precocity is legendary. At the age of 3 he was correcting his father's weekly wage calculations. When he was 7 he entered his first school, a squalid prison run by one Büttner, a brutal taskmaster. Two years later Gauss was admitted to the arithmetic class. Büttner had the endearing habit of giving out long problems of the kind, such as summing progressions, where the answer could readily be obtained from a formula—a formula known of course to the teacher, but not to the pupils. Each boy, on completing his task, had to place his slate on the master's desk. On one occasion no sooner had Büttner dictated the last number than his youngest pupil flung his slate on the desk and waited for an hour while the other boys toiled. When Büttner looked at Gauss' slate, he found there a single number—no calculation at all. Gauss liked to recall this incident in his later years, and to point out that his was the only correct answer.

We shall start with an arithmetic progression whose first term and common difference are 1. This is the progression

1, 2, 3,..., n

There is an anecdote about Carl Friedrich Gauss (1777–1855) that allegedly refers to this arithmetic progression in the case n = 100:

According to the tradition in the schools at that time, when a mathematics problem was given to a class, the pupil who finished first placed his slate board down in the middle of a large table, and then the next to finish put his slate down on top of it. One day, when young Carl was a pupil in Mr. Büttner's arithmetic class, Mr. Büttner posed the problem of adding an arithmetic progression. He had barely finished describing this task when Gauss threw his slate board on the table saying, in low Brunswick dialect, "Ligget se" ("there she lies"). While the other pupils continued to work on this problem, Mr. Büttner, conscious of his dignity, walked up and down the room, and occasionally threw a contemptuous and caustic glance at the smallest of his pupils, who had finished the task too quickly. At last the other slates began to come in; and when the slates were turned over, Mr. Büttner found that Gauss' solution was correct even though many of the others were wrong (and were corrected with a slapping). (Waltershausen [1856])

We can surmise that little Gauss had reasoned in the following way: We want to know the value of

S = 1 + 2 + ... 100.

S = 100 + 99 + ... 1.

2S = 101 + 101 + ... 101  =  100 × 101

S = (100 × 101)/2  = 5050.

Sn = 1 + 2 + ... + n.

Sn = n + (n-1) + ... + 1.

2Sn = (n+1) + (n+1) + ... + (n+1) = n(n+1),

Sn = n(n+1)/2.

This sounds like the story (recounted by Eric Temple Bell) about K. F. Gauss at the age of about 8 years, except that probably nobody considered Gauss to be "dull", just not yet at that age a great mathematician.

As I recall the story, Herr Büttner, the teacher, had given the boys in the class about an hour to add up a set of 100 numbers such as 5192 + 5229 + 5266 + ... , where each one was 37 larger than the previous one. I don't know the starting number nor the increment, but they formed an arithmetic progression, the kids were probably supposed to derive each term before adding it, and the teacher had a secret formula for determining the answer.

My guess is that Gauss figured out that the teacher had access to something he wasn't sharing and independently derived a slick way to find the sum, by rearranging the order of summing. Maybe it wasn't exactly divine inspiration, but it still took a pretty impressive mind to come up with that technique at that age.

Gauss just wrote the answer on his slate (no calculations), and he and Büttner sort of glared at each other for an hour while the other boys slaved away. Gauss later said that his answer was the only correct one turned in that day.

The story has a happy ending -- the teacher, recognizing that there wasn't much more that he could teach this unusual student, arranged for a tutor to take charge of Gauss's education, and the tutor and Gauss became lifelong friends and collaborators.

In order to savor once more this all too fugitive experience, here is a very different way of seeing that

1 + 2 + 3 + ... + n = n(n+1)/2.

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10

      ____________________________________
     |        ___________________         |
     |       |        ___        |        |
     |       |       |   |       |        |
     1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10
         |       |___________|       |
         |___________________________|
                  

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 10/2 × 11,

Some people relish the geometric approach, some of the symbolic. This tells you at once that personality plays as central a role in mathematics as in any of the arts. Proofs—those minimalist structures that end up on display in glass cases—come from people mulling things over in strikingly different ways, with different leapings and lingerings. But is it always from the same premises that we explore? Is there some sort of common sense that is to reason what Jung's collective unconscious used to be to the psyche? One of these approaches, or some third, must have been in the mind of the ten-year-old Gauss—the Mozart of mathematics—when, in his first arithmetic class, he so startled his teacher. It was 1787 and Herr Büttner was in the habit of handing out brutally long sums like these, which the children had to labor over. When each one finished, he added his slate to the pile growing on the teacher's desk. The morning might well be over before all had finished. But Gauss no sooner heard the problem than he wrote a single number on his slate and banged it down. "Ligget se'!" he said, in his Braunschweig accent: "There it lies!" And there it lay, the only correct answer in the lot.

Gauss was born into a family that, like many others of the time, had recently moved into town, hoping to improve its lot from that of impoverished farm workers. One of the benefits of living in Brunswick was that the young Carl could attend school. There are many stories told about Gauss's early-developing genius, one of which comes from his mathematics class when he was 9. At the beginning of the year, to keep his 100 pupils occupied, the teacher, J. G. Büttner, assigned them the task of summing the first 100 integers. He had barely finished explaining the assignment when Gauss wrote the single number 5050 on his slate and deposited it on the teacher's desk. Gauss had noticed that the sum in question was simply 50 times the sum 101 of the various pairs 1 and 100, 2 and 99, 3 and 98,... and had performed the required multiplication in his head.

Und dann gab er ihm einen Grund.

Büttner hatte ihnen aufgetragen, alle Zahlen von eins bis hundert zuzammenzuzählen. Das würde Studen dauern, und es war beim besten Willen nicht zu schaffen, ohne irgendwann einen Additionsfehler zu machen, für den man bestraft werden konnte. Na los, hatte Büttner gerufen, keine Maulaffen feilhalten, aufangen, los! Später hätte Gauß nicht mehr sagen können, ob er an diesem Tag müder gewesen war als sonst oder einfach nur gedankenlos. Jedenfalls hatte er sich nicht unter Kontrolle gehabt und stand nach drei Minuten mit seiner Schiefertafel, auf die nur eine einzige Zeile geschrieben war, vor dem Lehrerpult.

So, sagte Büttner und griff nach dem Stock. Sein Blick fiel auf des Ergebnis, und seine Hand erstarrte. Er fragte, was das solle.

Fünftausendfünfzig.

Was?

Gauß versagte die Stimme, er räusperte sich, er schwitzte. Er wünschte nur, er wäre noch auf seinem Platz und rechnete wie die anderen, die mit gesenktem Kopf dasaßen und taten, als hörten sie nicht zu. Darum sei es doch gegangen, eine Addition aller Zahlen von eins bis hundert. Hundert und eins ergebe hunderteins. Neunundneunzig und zwei ergebe hunderteins. Achtundneunzig und drei ergebe hunderteins. Immer hunderteins. Das könne man fünfzigmal machen. Also fünfzig mal hunderteins.

Büttner schweig.

Fünftausendfünfzig, wiederholte Gauß, in der Hoffnung, daß Büttner es ausnahmsweise verstehen würde.

100 +  99 +  98 + ... +   3  +  2 +   1
  1 +   2 +   3 + ... +  98  + 99 + 100
101 + 101 + 101 + ... + 101 + 101 + 101

The great mathematicians feel mathematics in a way the rest of us do not. And their genius for mathematics is immediately recognizable. When Gauss was eight years old, he and his classmates were asked by their teacher to find the sum of the integers from 1 to 100. The children began laboriously to calculate on their slates. All of them (except Gauss) began 1+2=3, 3+3=6, 6+4=10.... Gauss noticed that the integers 1, 2, 3,..., 99, 100 can be placed in pairs as follows: (1, 100), (2, 99), (3, 98), ..., (50, 51). There are exactly 50 such pairs and the sum of the integers in each pair is 101. Hence, the desired sum is the same as 50 times 101, which is 5050. Gauss wrote this number on his slate and handed it to the teacher. The whole process took him only seconds.

There is a well known story, repeated, with his usual trimmings, by Bell in his Men of Mathematics, that when Gauss was ten his teacher, Bütner, seeking an hour's repose, set his pupils the 100 term sum

81297 + 81495 + 81693 + ... + 100899

... Gauss flung his slate on the table: "There it lies," he said—"Ligget se'" in his peasant dialect. Then, for the ensuing hour, while the other boys toiled, he sat with his hands folded, favoured now and then by a sarcastic glance from Bütner, who imagined the youngest pupil in the class was just another blockhead. At the end of the period Bütner looked over the slates. On Gauss's there appeared but a single number. To the end of his days Gauss loved to tell how the one number he had written was the correct answer and how all the others were wrong.

   1+2+3+4+5+6+7 = (1+7) + (2+6) + (3+5) + 4
                 = (4+4) + (4+4) + (4+4) + 4
                 = 7 × 4 = 28

   1+2+3+4+5+6   = (1+6) + (2+5) + (3+4)
                 = (7/2 + 7/2) + (7/2 + 7/2) + (7/2 + 7/2)
                 = 6 × 7/2 = 21

a + (a+d) + (a+2d) + ... + (a+(n-1)d) = na + 1/2 n (n-1)d.

81297 + 81495 + 81693 + ... + 100899.

The number of risk parameters in a portfolio equals the sum of the number of assets it includes. For example, 5,050, the total number of volatilities and correlations for a 100 asset portfolio equals 1 + 2 + 3 + ... + 100.

There is an amusing and perhaps apocryphal story about this result and the famous mathematician Carl Friedrich Gauss, who was born in 1777 in Braunschweig, Germany. When Gauss was a child at St. Catherine elementary school, his teacher who was named Büttner asked the students in his class to sum the numbers from one to 100. Büttner's intent was to distract the students for a while so that he could tend to other business. To Büttner's surprise and annoyance, however, Gauss, after a few seconds, raised his hand and gave the answer—5,050. Büttner was obviously shocked at how quickly Gauss could add, but Gauss confessed that he had found a short cut. He described how he began by adding one plus two plus three but became bored and started adding backward from 100. He then noticed that one plus 100 equals 101, as does two plus 99 and three plus 98. He immediately realized that if he multiplied 100 by 101 and divided by 2, so as not to double count, he would arrive at the answer.

Young Gauss' formula gives the number of risk parameters for a portfolio comprising any number of assets—n times (n+1) divided by 2, where n represents the number of assets in the portfolio.

Nous sommes en 1786, comme chaque année le maître de la petite école de Brunswick commence son cours d'arithmétique par son exercice favori. Il demande à ses élèves de calculer la somme des 100 premiers entiers. À peine l'exercice posé, un de ses plus jeunes disciple, à peine agé de neuf ans, donne la réponse: 10100/2! Le maître qui avait l'habitude d'attendre un bon moment avant d'obtenir les premières réponses n'en croit pas ses oreilles et tombe de sa chaise! Il venait de découvrir l'un des plus grands génies mathématiques de tous les temps: Carl Friederich Gauss...

Arrivato all'età di dieci anni, viene dunque ammesso alle lezioni di aritmetica dell'autorità locale in materia: l'ormai dimenticato Buttner. Il professore ha fama di essere assai burbero e dai modi scostanti. Inoltre, pieno di pregiudizi fino al midollo, non ama gli allievi che provengono da famiglie povere, convinto che siano costituzionalmente inadeguati ad affrontare programmi culturali complessi e di un certo spessore. Il buon Buttner sarà costretto presto a ricredersi.

Un episodio in particolare viene ricordato nelle storie della matematica. Succede infatti che in una giornata particolare, in cui il professore aveva la luna più storta che in altre e in un momento in cui gli allievi si dimostrano più disattenti del solito, li obbliga, a mo' di esercizio punitivo, a calcolare la somma dei 100 primi numeri: 1+2+3+...+100. Proprio mentre comincia a gongolarsi al pensiero di quanto un suo trucchetto avrebbe lasciato a bocca aperta gli alunni, viene interrotto da Gauss che, in modo fulmineo afferma: "Il risultato è 5050". Rimane un mistero di come Gauss sia riuscito a realizzare la somma in maniera così rapida. Ad ogni modo, Buttner deve arrendersi di fronte all'enorme talento del giovane allievo e, con uno slancio che dopotutto lo riscatta di parecchio rispetto ai pregiudizi che aveva maturato, lo raccomanda al duca di Brunswick, supplicandolo di assicurare i mezzi economici sufficienti perché il genio in erba possa finire gli studi secondari e quelli universitari.

It was known in antiquity that if a₁, a₂, ... a_n are in arithmetic progression, then

a1 + a2 + ... + an = n(a1 + an)/2.

However it is one thing for a formula to be known by practicing mathematicians and quite another for it to be deduced in an instant by a ten-year-old boy. This is exactly what Gauss did when his arithmetic teacher, Herr Büttner, gave Gauss and his classmates a problem specifically designed to keep them hard at work for an hour. The problem chosen to create tedium and frustration was that of summing an arithmetic progression. Immediately Gauss wrote a number on his slate, turned it in, and announced, "There it is." At the end of the hour, the number written by Gauss was the only correct answer to come from the class. What Gauss immediately recognized was that in an arithmetic progression a₁, a₂, ... a_n,

a1 + an = a2 + an–1 = a3 + an–2 = ...,

After all these messages, I cannot resist telling what really happened, as I heard it from my high school teacher (he could compete with E. T. Bell for telling a good story). Gauss' teacher set the class the task of adding all the numbers from 1 to 100 on purpose to keep them busy for a long time, while the teacher would go to work at his vegetable garden, it was an urgent job. Gauss defeated his purpose by finding the answer instantly, so the teacher told the rest of the class to go on with the normal addition, and took Gauss with him to help dig out the potatoes.

Gauss began to show his prodigious mathematical talents at a very young age. He mastered the art of calculation before he could read or write, and at the age of three he supposedly found an error in his father's bookkeeping. There is also the famous story about the ten-year-old Gauss who, when asked by his teacher to find the sum of the integers from 1 to 100, almost instantly came up with the correct answer: 5,050. To the teacher's astonishment, Gauss explained that he had noticed that by writing the sum once as 1 + 2 + 3 + ... + 99 + 100, and again as 100 + 99 + ... + 3 + 2 + 1, and then adding the two lines, each pair of numbers added up to 101. Since there were 100 such pairs, the sum of the two rows was 100 × 101 or 10,100, and the sum of each row was one half of this, or 5,050.

Il primo episodio della vita di Gauss come matematico viene raccontato in tanti modi differenti, ma sostanzialmente simili; il maestro della scuola di Braunscweig, volendo passare un pomeriggio tranquillo, aveva assegnato un esercizio lungo e noioso, quello di sommare i numeri da uno a 80. Dopo pochi minuti, Gauss depose sulla cattedra la lavagnetta con il risultato, suscitando le ire del maestro che pensava a uno scherzo; tuttavia, un paio d'ore più tardi, quando tutti ebbero finito l'esercizio, dovette ricredersi, perché Gauss era uno dei pochi scolari che avevano trovato il risultato esatto. Stupito, il maestro chiese al ragazzo come fosse riuscito a calcolare tanto rapidamente e Gauss gli fece notare che i numeri si possono scrivere in sequenza ascendente o discendente così:

  1   2   3   4   5   6   7   8   9  ... 79 80
 80  79  78  77  76  75  74  73  72  ...  2  1

Without the help or knowledge of others, Gauss learned to calculate before he could talk. At the age of three, according to a well-authenticated story, he corrected an error in his father's wage calculations. He taught himself to read and must have continued arithmetical experimentation intensively, because in his first arithmetic class at the age of eight he astonished his teacher by instantly solving a busy-work problem: to find the sum of the first hundred integers. Fortunately, his father did not see the possibility of commercially exploiting the calculating prodigy, and his teacher had the insight to supply the boy with books and to encourage his continued intellectual development.

Before he could talk, Carl had learned to calculate, and at age three he had corrected mistakes in his father's wage calculations! In his eighth year, while in his first arithmetic class, Gauss found a formula for the sum of the first n consecutive numbers. His teacher, suitably impressed, supplied the boy with literature to encourage his intellectual development.

Carl Friedrich Gauss (1777–1855) is considered to be among the greatest mathematicians who ever lived. His genius was evident at the age of three when he corrected an error in his father's bookkeeping. Also, at the age of eight, he astonished his teacher, Büttner, by rapidly adding the integers from 1 to 100 via the observation that the fifty pairs (j+1, 100–j) for j = 0, 1, ..., 49 each sum to 101 for a total of 5050.

Even as a toddler Carl showed signs of genius, which his parents interpreted as indicating an early death, for God's favorites die young. Carl could add and subtract almost before he could talk. One day while his father added up a long row of figures, three-year-old Carl watched patiently and when the sum was written down, exclaimed, "Father, the answer is wrong. It should be–––." Gebhard Gauss redid the figures and discovered that his son was right—there was an error and the answer Carl had given was the correct one.

The little prodigy learned to read as mysteriously and easily as he had learned to add. He implored his father to teach him the alphabet and then, armed with this knowledge, went off and taught himself to read.

His precocious achievements were proudly displayed as though they were parlor tricks. Little Carl was popped into a chair and asked to add figures his father wrote on a slate while an audience of friends and relatives looked on admiringly. Unfortunately, Gauss inherited poor eyesight as well as genius and was unable to see the numbers. Too shy to admit it, he simply sat there while admiring looks turned to nods of "I thought so."

Parlor tricks are one thing, genius is another—and Carl's father was either unable or unwilling to recognize the latter in his son. He set him to spinning flax in the afternoons in order to supplement the family income, and had every expectation that Carl would learn a trade of some sort—perhaps weaving, like his uncle Johann Benze, whom Carl adored. It was Johann who first recognized and cultivated Carl's talents, evidently seeing in his nephew the hopes for all his own frustrated dreams.

At the age of seven, Carl was sent to the local grammar school, where the tyrant of a teacher thought nothing of using a whip to beat an education into the boys. To keep the class busy one day, he assigned them the problem of adding all the numbers from one through a hundred. When the pupils finished, they were supposed to lay their slates on the table in the front of the room. The teacher had no sooner stated the problem than Carl scribbled the answer on his slate and tossed it on the table saying, "Ligget se," low German for "There it is." No one had ever told Carl the formula for adding a sequence of numbers, and the teacher was astounded that he had discovered it for himself. It is the same formula that the Pythagoreans had used as a password in their secret society: 1/2n(n+1) = S, where S is the sum and n is the last number of the sequence 1, 2, 3 ... n. Gauss probably figured out the solution by adding 100 and 1, 99 and 2, 98 and 3, and so on. In each case the answer is 101, and since there are 100 numbers to be added, there are fifty sets of 101. Fifty times 101 is 5,050, the answer to the problem. (Or, by the formula, 1/2 × 100 × 101 = 5,050.)

Era una mañana común y corriente en una escuela como cualquier otra. El profesor, ante un grupo de niños de alrededor do 10 años de edad, estaba molesto por algún mal comportamiento del grupo y decidió poner a trabajar a sus alumnos en un problema de matemáticas que segun él les llevaría un buen rato terminar; así, de paso, podría descansar un poco. En esa época se acostumbraba que los niños llevaran una pequeña pizarra en la cual hacían sus ejercicios. El maestro dijo a sus alumnos que según fueran terminando el problema dejaran las pizarras boca abajo sobra su escritorio, para que terminar todos él revisara los resultados. El problema consistía en sumar los primeros cien números enteros, es decir, encontrar la suma de todos los numeros del 1 al 100.

A los pocos segundos de haber plamteado el problema, se levantó un niño y depositó su pizarra sobre el escritorio del maestro. Éste, convencido de que aquel niño no quería trabajar, ni se molestó en ver el resultado; prefirió esperar a que todos terminaran. Un poco más de media hora después comenzaron a levantarse los demás niños para dejar su pizarra, hasta que finalmente todo el grupo terminó.

Para sorpresa del profesor, de todos los resultados el único correcto era el del muchacho que había entragado primero. Mandó llamar al chico y le preguntó si estaba seguro de su resultado y cómo lo había encontrado tan rápido. El niño respondió: "Mire, maestro, antes de empazar a sumar mecánicamente los cien primeros números me di cuenta de que si sumaba el primero y el últimto obtenía 101; al sumar el segundo y el punúltimo también se obtiene 101, al igual que al sumar el tercero y el antepenúltimo, y así sucesivamente hasta llegar a los dos números centrales que son 50 y 51, que también suman 101. Entonces lo que hice fue multiplicar 101 por 50 para obtenir mi resultado do 5050."

At the age of seven, Carl Friedrich Gauss started elementary school, and his potential was noticed almost immediately. His teacher, Büttner, and his assistant, Martin Bartels, were amazed when Gauss summed the integers from 1 to 100 instantly by spotting that the sum was 50 pairs of numbers each pair summing to 101.

Carl Friedrich Gauss, possibly the greatest mathematician of all time, showed his arithmetical skill at an early age. When he was ten years old his class at school was given what was intended to be a long routine drill exercise by a tyrannical schoolmaster: "Find the sum of the first 100 positive integers." This was easy for the schoolmaster, who knew how to sum arithmetic progressions, but the formula was unknown to the boys. Young Gauss did not know how to do it either, but he invented a way, instantly and in his head. Writing the answer on his slate, he handed it in at once. When the rest of the students' calculations were collected an hour later, all were found to be incorrect except Gauss's! We are told that he did it by pairing the terms and then mentally multiplying the value of each pair by the number of pairs. If the pairs could each total 100, so much the easier: 100 + 0, 99 + 1, etc. This would make 50 pairs of 100 each for 5000, plus 50 left over (the middle number), for a total of 5050.

Herr Oberlehrer Büttner staunt Schon in seiner Jugend zeigte sich die mathematische Begabung von Carl Friedrich Gauß, der von sich sagte, er habe früher rechnen als sprechen können. Bereits im zarten Alter von drei Jahren soll er seinen Vater bei einem Fehler in einer Lohnabrechnung korrigiert haben. In der dritten Volksschulklasse, also im Alter von etwa neun Jahren, demonstrierte er seine herausragenden mathematischen Fähigkeiten auf eindrucksvolle Weise. Der Lehrer Johann Georg Büttner hatte der Klasse die Aufgabe gegeben, die Zahlen von 1 bis 100 zu addieren. Gauß löste diese Aufgabe auf schnelle und elegante Weise.

Anstatt alle hundert Zahlen zusammen zu zählen, bildete er Zahlenpaare: Bei der Addition der ersten (1) und der letzten Zahl (100) der Folge ergibt sich 101, wie auch bei der Addition der zweiten (2) und der vorletzten (99), der dritten (3) und der drittletzten (98) ... Insgesamt ergeben sich also 50 Zahlenpaare, die jeweils die Summe 101 ergeben. Mit diesen Überlegungen konnte Gauß die vom Lehrer gestellte Additionsaufgabe (1+2+...+99+100) in eine rechentechnisch weitaus einfachere Multiplikation (50×101) umwandeln. Gauß war der einzige Schüler, der die Aufgabe richtig löste – und er war mit seinen Berechnungen auch noch mit Abstand der Schnellste.

Mathematicians have always been fascinated by accounts of precocious mathematical achievements. They all know the story of Carl Friedrich Gauss, who was born in Brunswick, Germany, in 1777. When Gauss was three, his father was making out a weekly payroll when the little boy, peering over his shoulder, corrected his addition. When Gauss was ten, the teacher at his school decided to keep the students busy by having them add the numbers from 1 to 100. Gauss had never seen the problem before, but he immediately figured out a clever way to calculate the sum quickly. He wrote the answer on his slate, marched to the front of the room, and deposited the slate on the teacher's desk. Later in life Gauss liked to recount how his was the only correct answer, even though his classmates worked for hours laboriously adding number after number.

It is said that while Gauss was attending elementary school, his teacher had once given to the class the following exercise: add 2 to 1, then add 3 to the previous sum, and continue like this until you reach 100. The teacher expected that while the students were busy adding all those numbers, he could enjoy a peaceful break, long enough to digest his meal. But after only a few minutes, he noticed that Gauss had stopped calculating. Intrigued, he went to check the child's copybook and found that, after a few additions, Gauss had multiplied 100 by 101 and then divided the product by 2, obtaining 5,050, which is the right answer. If he had relied on axiom 5, Gauss might have remarked that 1+2=3, 1+2+3=6, 1+2+3+4=10, and that if the last number added is n, then the sum equals n(n+1)/2. Hence his simple calculation of the correct answer.

[footnote] As a matter of fact, young Gauss added 1 to 100 and found 101, and the same result for 2 + 99, and so on. Then he only had to multiply 101 by the number of such partial sums, namely, 50.

The year was 1787. Ten year old Carl Friedrich Gauss was enrolled at a primary school in Germany. Although his teacher did not think so, Carl was a very bright student. At times his attention wandered, but he loved to learn and discover new ideas, especially in mathematics. Carl's teacher, Master Büttner, was a good teacher of history and Latin, but he did not like to teach mathematics. He spent little time on the subject, and had the students do tedious problems or problems which they already understood. Today's math lesson was no different. Little did Master Büttner know he would be in for a surprise.

He walked up to the chalk board to write the day's problem. "Students, for math work today, I want you to add the whole numbers from 1 to 100." And he wrote—"Add the whole numbers from 1 to 100."

"Get busy," he said.

Each student had a slate board and chalk to work on, and by tradition the first student finished would put his slate face down on Master Büttner's table. As each student completed the task, his slate would join the pile. All the students except Carl unenthusiastically pulled out their slates, and began adding. Carl sat at his desk with his hands on his chin, thinking about the problem.

Master Büttner saw him and thought Carl was daydreaming. He shouted, "Get busy, Carl Gauss." Startled, Carl looked up and said, "I already have the answer, Master Büttner."

"What do you mean you have the answer? You have not written anything on your slate," retorted Master Grumple.

"I did the problem in my head," replied Carl. Master Büttner got a nasty smile on his face and said, "Well then, come up to the board and show us all how you solved it, Carl." Master Büttner thought he had trapped Carl in a lie. Carl stood up and walked slowly but confidently to the front of the room. He went to the chalk board and began to write and explain the following:

     1 + 2 + 3 + 4 + ... + 97 + 98 + 99 + 100
     |   |   |   |_________|    |    |     |
     |   |   |__________________|    |     |
     |   |___________________________|     |
     |_____________________________________|

"Each pair totaled 101, and I figured there were 50 of these 101s. So I multiplied 101 × 50 and got 5050 for the sum of 1 + 2 + 3 + 4 + 5 + ... + 95 + 96 + 97 + 98 + 99 + 100."

Everyone was startled, especially Master Büttner. "The boy is right," he thought. All the students applauded, and Master Büttner had to compliment Carl on his work.

Carl Frederich Gauss went on to become a famous mathematician.

One of the great mathematicians of all times was Karl Friedrich Gauss. He took his first arithmetic class when he was seven years old. The teacher, a man named Büttner, loved to make life miserable for his students. He would even thrash them at every opportunity. This is a story Gauss himself liked to tell in his old age. Of course, he may have embellished it a bit over the years.

One day Büttner decided he would keep the students busy while he would attend to his own interests. He told the students to add all the numbers fro