Versions of the Gauss Schoolroom Anecdote

Transcribed below are 109 tellings of the story about Carl Friedrich Gauss's boyhood discovery of the "trick" for summing an arithmetic progression.

A word about how I collected these accounts:

Many were found through conventional methods of library research. I began with biographies of Gauss, then followed references mentioned by the biographers, and I was also guided by the major Gauss bibliography assembled by Uta C. Merzbach (Merzbach, Uta C. 1984. Carl Friedrich Gauss: a Bibliography. Wilmington, Dela.: Scholarly Resources.) To broaden the search I thumbed through various well-known works on the history of mathematics and collections of mathematical anecdotes, and I browsed in literature on the teaching of mathematics.

On the World Wide Web, search engines offered a very efficient means of locating versions of the story. I tried various combinations of search terms such as "Gauss," "Büttner," "slates," and "progression" (or their equivalents in other languages). Another invaluable resource was the Google Book Search. This service has been controversial because some authors and publishers maintain it infringes their copyrights. Whatever the outcome of that dispute, I can report that Google Book Search led me to many works I would never have found by any other means. I would not have thought to look for the Gauss story in PHP and PostgreSQL: Advanced Web Programming or in a book titled Puzzles of Finance: Six Practical Problems and their Remarkable Solutions.

My thanks to the librarians of the following institutions: Boston College, the Boston Public Library, Boston University, Duke University, Mt. Holyoke College, Johns Hopkins University, the Library of Congress, the Massachusetts Institute of Technology, the University of North Carolina, North Carolina State University, Northwestern State University in Natchitoches, Mississippi, and the Wake County (North Carolina) Public Library. Especial thanks to Carolina Grey at Johns Hopkins and Mary Linn Wernet in Natchitoches.

Johannes Berg of the University of Cologne and Stephan Mertens of the University of Magdeburg helped me in this curious pursuit by supplying documents I could not obtain in the U.S. Margaret Tent of the Altamont School in Birmingham, Alabama, shared passages from her new biography of Gauss (The Prince of Mathematics: Carl Friedrich Gauss) in advance of publication. Ivo Schneider of the Bundeswehr University, Munich, offered advice on interpreting the documentary record of Gauss's early life (but obviously he is not to be held responsible for my interpretations).

The versions of the tale presented here are only a sample of those in the worldwide literature. I would be happy to receive other tellings of the story, in any language, and will attempt to include them in this archive. Of particular interest are any versions that predate those of Eric Temple Bell and Ludwig Bieberbach in 1937 and 1938. Please send all such materials, and any corrections of the transcriptions found here, to bhayes@amsci.org.

Alamo, Fernando de. 2005. Gauss y la campana. Weblog posting in Historias de la Ciencia, 22 December 2005. Link to Web page (Viewed 2006-02-03)

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.

Bell, E. T. 1937. Men of Mathematics. New York: Simon and Schuster. (See chapter 14, "The Prince of Mathematicians: Gauss," pp. 218–269.)

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.

Bieberbach, Ludwig. 1938. Carl Friedrich Gauß: Ein Deutsches Gelehrtenleben. Berlin: Keil Verlag. (pp. 14–15}

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.

Borwein, Jonathan, and David H. Bailey. 2004. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Natick, Mass.: A K Peters. (p. 42)

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!

Bos, H. J. M. 1978. Gauss 1777–1855. In Carl Friedrich Gauss, 1777-1855: Four Lectures on his Life and Work. Edited by A. F. Monna. Utrecht: Communications of the Mathematical Institute, Rijksuniversiteit Utrecht. (p. 4)

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.

Boutiche, Saïd. 2005. Représentation mentale, cognition et mathématiques. Link to Web page

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
ou bien si l'on fait l'arrangement spatial suivant:

         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.

Boyer, Carl B. 1968, 1991. A History of Mathematics. Second edition. Revised by Uta C. Merzbach; foreword by Isaac Asimov. New York: Wiley. (p. 497)

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....

Bruce, Donald, and Anthony Purdy (editors). 1994. Literature and Science. Amsterdam: Editions Rodopi. (pp. 73–74)

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.

Burkholder, Peter J. 1993. Alcuin of York's Propositiones ad acuendos juvenes: Introduction, commentary and translation. HOST: An Electronic Bulletin for the History and Philosophy of Science and Technology, Vol. 1, No. 2. Link to Web page

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.

Burrell, Brian. 1998. Merriam-Webster's Guide to Everyday Math: A Home and Business Reference. Springfield, Mass.: Merriam-Webster. (p. 24)

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.

Burton, David M. 1976. Elementary Number Theory. Boston: Allyn and Bacon. (pp. 68–69)

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+100=101, 2+99=101, 3+98=101, ..., 50+51=101.

Since there are 50 pairs of numbers, each of which adds up to 101, the sum of all the numbers must be 50 × 101 = 5050. This technique provides another way of deriving the formula

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

for the sum of the first n positive integers. One need only display the consecutive integers 1 through n in two rows as follows:
   1    2   3  ... n-1  n
   n  n-1 n-2  ...   2  1
Addition of the vertical columns produces n terms, each of which is equal to n+1; when these terms are added, we get the value n(n+1). Because the same sum is obtained on adding the two rows horizontally, what occurs is the formula n(n+1) = 2(1 + 2 + 3 + ... + n).

Cantor, Moritz. 1878. Carl Friedrich Gauss. In Allgemeine Deutsche Biographie Vol. 8, pp. 430–445. Leipzig: Verlag von Duncker & Humblot.

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.

Choi, Young Back. 1993. Paradigms and Conventions: Uncertainty, Decision Making, and Entrepreneurship. Ann Arbor: University of Michigan Press. (p. 33)

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.

Daepp, Ulrich, and Pamela Gorkin. 2003. Reading, Writing, and Proving: A Closer Look at Mathematics. New York: Springer Science+Business Media. (p. 209.)

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
Now add vertically. When you do this, you will get 101 one hundred times; in other words, you get (101)(100). This is twice the sum that you needed, so the answer must be (101)(100)/2. There is nothing special about the integer 100. If you try this with a general positive integer n, you will see that 1 + 2 + 3 + ... + n = n(n+1)/2 for every positive integer n. What a nice formula! Is something like this true for the sums of squares of the first n integers? Indeed it is. We'll give it a rigorous proof using mathematical induction.

Dartmouth. 2000. Posting on open forum for class Soc 15, Quantitative Analysis of Social Data. Dartmouth College. (Header reads: "single on Tue Jun 6 09:32:47 2000.") Link to Web page

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.

Dehaene, Stanislas. 1997. The Number Sense: How the Mind Creates Mathematics. New York: Oxford University Press. (pp. 147–148)

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.

Derbyshire, John. 2003. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. Washington, D.C.: Joseph Henry Press. (pp. 48–49)

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.

Devlin, Keith. 1997. Mathematics: The Science of Patterns: The Search for Order in Life, Mind, and the Universe. New York: Scientific American Library. (p. 21)

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.
Now you add the two sums, column by column, to give
101 + 101 + 101 + ... + 101 + 101 + 101.
There are exactly 100 copies of the number 101 in this sum, so its value is
100 × 101 = 10,100.
Since this product represents twice the answer to the original sum, if you halve it you obtain the answer Gauss' teacher was looking for, namely 5050.

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.
This formula gives the general pattern of which Gauss' observation was a special case.

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.

Dunham, William. 1990. Journey through Genius: The Great Theorems of Mathematics. New York: Wiley. (pp. 236–237)

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.
Instead of adding these numbers horizontally across the rows, Gauss added them vertically down the columns. In so doing, of course, he got
2S = 101 + 101 + 101 + ... +  101 + 101 + 101
since the sum of each column is just 101. But there are a hundred columns. Thus 2S = 100 × 101 = 10100, and so the sum of the first hundred whole numbers is just
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.

Dunnington, G. Waldo. 1955. Carl Friedrich Gauss: Titan of Science. Reprinted 2004 with additional material by Jeremy Gray and Fritz-Egbert Dohse. Washington, DC: Mathematical Association of America.

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.

Du Sautoy, Marcus. 2003. The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics. New York: Harper Collins. (pp. 24–25)

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.

Elta Universitate. Undated web site. Carl Friedrich Gauss. Link to Web page (Viewed 2006-03-11)

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.

Estep, Donald. 2002. Practical Analysis in One Variable. New York: Springer-Verlag. (pp. 29–30)

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)
Note that the "..." indicates that we add all the natural numbers between 1 and n. This formula makes it possible to replace the n-1 additions on the left by a multiplication and a division, which is a considerable reduction in work when using a piece of chalk and a slate to do the sums.

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
This agrees with the formula (3.1) with n = 99. In Problem 3.9, we ask you to show that this argument can be used to show that (3.1) holds for any natural number n.

Eves, Howard W. 1969. In Mathematical Circles: A Selection of Mathematical Stories and Anecdotes. Vol. 2, Quadrants III and IV. Boston: Prindle, Weber & Schmidt. (p. 112, item 319)

319° Gauss's precocity.

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."

Fadiman, Clifton, and Andre Bernard, general editors. 1985, 2000. Bartlett's Book of Anecdotes. Revised edition. Boston: Little, Brown and Company. (p. 229)

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.

Falbo, Clement. 2000. Math Odyssey. Champlain, Ill.: Stipe Publishing Co. Excerpts available online. Link to Web page

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.

Fong, Peter. 2003. SIMMS Integrated Mathematics. A Modeling Approach Using Technology: Integrated Mathematics, Level 2. Student edition. Dubuque, Ia.: Kendall/Hunt Publishing Company. (pp. 381–382)

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
This series can also be written in reverse order, as shown below.
S100 = 100 + 99 + ... + 2 + 1
These two series can then be added as follows:
 S100 =     1  +    2  +  ...  +   99  +  100
 S100 =   100  +   99  +  ...  +    2  +    1
2S100 =   101  +  101  +  ...  +  101  +  101
Since the resulting equation contains 100 terms of 101, the sum of the two equations can be written as:
2S100 = 100(101)
Solving the equation for S100, the sum of the first 100 natural numbers can be found as follows:
S100 = (100(101))/2 = 5050

Friendly, Michael. 1988. Advanced LOGO: A Language for Learning. Hillsdale, N.J.: Lawrence Erlbaum Associates. (pp. 64–66)

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.

Galle, A. 1916. C. F. Gauss als Zahlenrechner. In Matierialien für eine wissenschaftliche Biographie von Gauss, compiled by F. Klein, M. Brendel and L. Schlesinger. Vol. IV. Leipzig: B. G. Teubner. Available online. Link to PDF file (pp. 4–5)

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.

Gauss2005 web site. 2005. Carl Friedrich Gauss (1777–1855) — sein Leben. Link to Web page (Viewed 2005-11-25)

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.

Geschwinde, Ewald, and Hans-Jürgen Schönig. 2002. PHP and PostgreSQL: Advanced Web Programming. Indianapolis, Ind.: Sams Publishing. (p. 89)

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);
      }
}
?>

Giancoli, Douglas C. 2000. Physics for Scientists and Engineers with Modern Physics. Third edition. Upper Saddle River, N.J.: Prentice Hall. Available online. Link to Web page (Viewed 2006-02-02)

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.

Gindikin, S. 1999, 2000. Carl Friedrich Gauss. Quantum 10(2):14–19 and 10(3):10–15. (Vol 10(2), p. 14)

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.

Goldman, Phyllis (editor). 2002. Monkeyshines Explores Math, Money, and Banking. Greensboro, N.C.: The North Carolina Learning Institute for Fitness and Education. (p. 57)

Gauss (1777–1855). The Prince of Mathematics.

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.

Goldstein, Martin, and Inge F. Goldstein. 1984. The Experience of Science: An Interdisciplinary Approach. New York: Plenum Press. (p. 89)

... 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.

Gosselin, Marie-Ève. 2000. Gauss et le GAUS. L'Attracteur: La Revue de Physique. No. 9, Hiver 2000. Available online. Link to Web page (Viewed 2006-02-15)

Né à Braunschweig (Allemagne), le 30 avril 1777, Gauss montra, dès son jeune âge, des aptitudes hors du commun pour les mathématiques (il faut mentionner qu'il avait appris à lire et à compter, par lui-même, à l'âge de 3 ans!). Un jour, à l'école primaire, un professeur demanda à ses élèves d'écrire tous les chiffres de 1 à 100 et d'en calculer la somme. Quelques temps à peine après cette consigne, le petit Gauss alla voir son professeur et lui montra sa réponse: 5050, ce qui était exact! Le professeur, stupéfait, demanda à l'enfant ce qu'il avait fait pour trouver le bon résultat si rapidement. Le prodige avait remarqué que 1 + 2 + 3 + ... + 99 + 100 équivalait à (1 + 100) + (2 + 99) + ... + (49 + 52) + (50 + 51). La somme de chacune de ces paires — il y en avait 50 — était 101. Il avait donc trouvé la réponse finale en calculant 50 × 101, soit 5050! On peut généraliser le raisonnement de Gauss par la formule suivante, qui permet de faire rapidement l'addition d'une suite de nombres:
somme = (n(n+1))/2

Gowar, Norman. 1979. An Invitation to Mathematics. Oxford: Oxford University Press. (pp. 7–11)

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.

Graham, Ronald L., Donald E. Knuth and Oren Patashnik. 1989. Concrete Mathematics. Reading, Mass.: Addison-Wesley Publishing Company. (p. 6)

To evaluate Sn [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)
We merely add Sn to its reversal, so that each of the n columns on the right sums to n+1. Simplifying, Sn = n(n+1)/2, for n≥0

Graham, Ronald L., Donald E. Knuth and Oren Patashnik. 1989. Concrete Mathematics. Reading, Mass.: Addison-Wesley Publishing Company. (p. 30)

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).
By the cummutative law we can replace k by n-k, obtaining
  S = Σ(a+b(n-k)) = Σ(a+bn-bk).
These two equations can be added by using the associative law:
  2S = Σ((a+bk)+(a+bn-bk)) = Σ(2a+bn).
And we can now apply the distributive law and evaluate a trivial sum:
  2S = (2a+bn)Σ 1 = (2a+bn)(n+1).
Dividing by 2, we have proved that
  Σ(a+bk) = (a + 1/2bn)(n+1).
The right-hand side can be remembered as the average of the first and last terms, namely 1/2(a+(a+bn)), times the number of terms, namely (n+1).

Grégory. 2005. Gauss, l'enfant prodige. Blog posting, Jeudi 14 juillet 2005. Link to Web page (Viewed 2006-02-15)

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!

Hall, Tord. 1970. Carl Friedrich Gauss: A Biography. Translated by Albert Froderberg. Cambridge: M.I.T. Press. (pp. 3–5)

The Mathematical Prodigy.

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.


Schooling.

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
Addition in the vertical columns gives 100 terms, each of which is equal to 101. Since this is twice the sum wanted, the answer is 50×101=5050.

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.

Hannoversch Münden web site. Undated. No author listed. Carl Friedrich Gauss. Link to Web page

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).

Hänselmann, Ludwig. 1878. Karl Friedrich Gauß: Zwolf Kapitel aus Seinem Leben. Leipzig: Duncker and Humblot. (pp. 16–17)

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.

Hartmann, Caroline. 1997. Carl Friedrich Gauß und die Geometria Situs des Universums. Ibykus No. 58 (1/1997). Link to Web page (Viewed 2006-02-15)

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.

Hein, James L. 2002. Discrete Structures, Logic, and Computability. Sudbury, Mass.: Jones and Bartlett. (pp. 256–257)

A Classic Example: Arithmetic Progressions

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
The trick is to notice that the sum of the first and last numbers, which is 30, is the same as the sum of the second and next to last numbers, and so on. In other words, if we list the numbers in reverse order under the original list, each column totals to 30.
   3    7   11   15   19   23   27
  27   23   19   15   11    7    3
  30   30   30   30   30   30   30
If S is the sum of the progression, then 2S=7(30). So S = 105.

The Sum of an Arithmetic Progression

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

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

Hoffman, Paul. 1998. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion. (pp. 171–173)

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
Gauss realized that you could add the numbers vertically instead of horizontally. There are 100 vertical pairs, each summing to 101. So the answer is 100 times 101 divided by 2, since each number is counted twice. Gauss easily did the arithmetic in his head.

"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."

Hollingdale, Stuart. 1977. C. F. Gauss (1777–1855): A bicentennial tribute. Bulletin Institute of Mathematics and its Applications 13(3–4):68–76. Reprinted in Makers of Mathematics, 1989, London: Penguin Books. (p. 314)

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.

Jacobs, Konrad. 1992. Invitation to Mathematics. Princeton: Princeton University Press. (pp. 72–74)

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.
Reversing the order of the terms, we can also write this number as
S = 100 + 99 + ... 1.
Adding the terms that lie in the same vertical line we obtain
2S = 101 + 101 + ... 101  =  100 × 101
and so
S = (100 × 101)/2  = 5050.
Therefore the number that Gauss wrote on his slate should have been 5050. The method we have just described for summing an arithmetic progression is both fast and simple, and because it is simple, it is not prone to computational errors. We shall now repeat the method to obtain the more general sum
Sn = 1 + 2 + ... + n.
Reversing the order of the terms we obtain
Sn = n + (n-1) + ... + 1.
Therefore
2Sn = (n+1) + (n+1) + ... + (n+1) = n(n+1),
and so we have
Sn = n(n+1)/2.

Johns, Vincent. 1997. Usenet posting in news group alt.algebra.help, May 18, 1997, in thread "sum of sequence of 1/n." Link to Web page (Viewed 2006-02-18)

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.

Kaplan, Robert, and Ellen Kaplan. 2003. The Art of the Infinite: The Pleasures of Mathematics. New York: Oxford University Press. (pp. 30–31)

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.
Again we choose an example—say, 10. You look at the sum
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10
and ingrained habits of reading from left to right, as well as being systematic, lead you to starting: 1 plus 2 is 3, and 3 makes 6, and 4 makes 10... But what if you look at it differently (and the seccret of all mathematical invention is looking from an unusual angle)—what if you add in pairs as follows:
      ____________________________________
     |        ___________________         |
     |       |        ___        |        |
     |       |       |   |       |        |
     1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10
         |       |___________|       |
         |___________________________|
                  
1+10=11, 2+9=11, 3+8=11—in fact, all these pairs will add up to 11! And how many pairs are there? 5—that is, half of 10. So
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 10/2 × 11,
or (n/2)(n+1).

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.

Katz, Victor J. 1998. A History of Mathematics: An Introduction. Second edition. Reading, Mass.: Addison-Wesley. (p. 654)

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.

Kehlmann, Daniel. 2005. Die Vermessung der Welt. Hamburg: Rowohlt, Reinbek. (p. 56)

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.

Kilpatrick, Jeremy, Jane Swafford and Bradford Findell (editors). 2001. Adding It Up: Helping Children Learn Mathematics. Washington, D.C.: National Academy Press. (pp. 108–109)

A closely related numerical approach to the problem of counting handshakes comes from a story told of young Carl Friedrich Gauss (1777–1855), whose teacher is said to have asked the class to sum the numbers from 1 to 100, expecting that the task would keep the class busy for some time. The story goes that almost before the teacher could turn around, Gauss handed in his slate with the correct answer. He had quickly noticed that if the numbers to be added are written out and then written again below but in the opposite order, the combined (double) sum may be computed easily by first adding the pairs of numbers vertically and then adding horizontally. As can be seen below, each vertical sum is 101, and there are exactly 100 of them. So the double sum is 100 × 101, or 10,100, which means that the desired sum is half that, or 5050.
100 +  99 +  98 + ... +   3  +  2 +   1
  1 +   2 +   3 + ... +  98  + 99 + 100
101 + 101 + 101 + ... + 101 + 101 + 101

King, Jerry P. 1992. The Art of Mathematics. New York: Plenum Press. (p. 40)

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.

Körner, T. W. 1996. The Pleasures of Counting. Cambridge University Press. (p. 281)

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
The teacher had barely finished stating the problem when, to quote Bell:
... 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.
(A more restrained account of Gauss's early life, and a more sympathetic estimate of Bütner will be found in Bühler's excellent biography.)

(i) Verify that
   1+2+3+4+5+6+7 = (1+7) + (2+6) + (3+5) + 4
                 = (4+4) + (4+4) + (4+4) + 4
                 = 7 × 4 = 28
and
   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

(ii) Work out Bütner's sum.

(iii) Show more generally that
a + (a+d) + (a+2d) + ... + (a+(n-1)d) = na + 1/2 n (n-1)d.

Krantz, Steven G. 2005. Mathematical Apocrypha Redux: More Stories and Anecdoetes of Mathematicians and the Mathematical. Washington, D.C.: Mathematical Associaiton of America.

It is well known among mathematicians—a story that we all tell to our calculus students—that Carl Friedrich Gauss, when he was ten years old—stunned his schoolteacher by performing the sum 1 + 2 + 3 + ... + 99 + 100—which the teacher had given the class in order to fill up the afternoon—in a minute or two. What Gauss did was to observe that the sum of an arithmetic series is the number of terms multiplied times the average of the first and last term. The story has, however, been transmogrified with time. It is thought that the actual sum that Gauss was asked to calculate was

81297 + 81495 + 81693 + ... + 100899.
This sum can of course be calculated by the same method.

Kritzman, Mark P. 2000. Puzzles of Finance: Six Practical Problems and their Remarkable Solutions. New York: John Wiley and Sons. (p. 118)

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.

Langevin, Philippe. 1999. Quelques Propriétés Cachées des Nombres. Link to Web page (Viewed 2006-02-15)

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...

Leonardi.it Web site. 2003. Johann Gauss: Dare i numeri fa bene. Link to Web page (Viewed 2006-02-15)

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.

Lozansky, Edward, and Cecil Rousseau. 1996. Winning Solutions. New York: Springer-Verlag. (p. 46)

It was known in antiquity that if a1, a2, ... an 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, ... an,

a1 + an = a2 + an–1 = a3 + an–2 = ...,
so the sum is the same as if every one of the n terms had the "average" value (a1+ an)/2.

Mann, Avinoam. 1998. Re: Re[2]: C. F. Gauss, Boy Wonder: speculation on what really happened. Posting to Discussion List on the History of Mathematics, Tue, 17 Nov 1998 03:27:56 +0200. Link to Web page (Viewed 2006-02-15)

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.

Maor, Eli. 1991. To Infinity and Beyond: A Cultural History of the Infinite. Princeton, N.J.: Princeton University Press. (p. 123)

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.

Maxint. Undated Web site. Galleria dei grandi matematici della storia. Link to Web page (Viewed 2006-02-18)

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
e che la somma di ogni coppia di numeri in colonna è sempre ottantuno; basta quindi moltiplicare ottantuno per le ottanta coppie e dividere per due per ottenere 3240, cioè il risultato dell'esercizio.

May, Kenneth O. 1972. Carl Friedrich Gauss. Dictionary of Scientific Biography (Vol. 5, pp. 298–315). New York: Scribner. (p. 298)

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.

McElroy, Tucker. 2005. A to Z of Mathematicians. New York: Facts on File, Inc. (pp. 112–115)

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.

Mollin, Richard A. 1999. Algebraic Number Theory. Boca Raton, Fla.: CRC Press. (footnote, p. 26)

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.

Muir, Jane. 1961. Of Men and Numbers: The Story of the Great Mathematicians. New York: Dodd, Mead. (Reprinted by Dover Publications, 1996) (p. 158)

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.)

Noreña, Francisco. 1992. El develador de las incógnitas: Carl Friedrich Gauss. México: Pangea. (pp. 12–14)

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."

O'Connor, J. J., and E. F. Robertson. 1996. Johann Carl Friedrich Gauss. MacTutor. Link to Web page (Viewed 2005-11-24)

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.

Ogilvy, C. Stanley, and John T. Anderson. 1966. Excursions in Number Theory. New York: Oxford University Press. (Reprinted 1988 Dover Publications.) (p. 12)

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.

Ohm, Matthias. 2005. Wie der Blitz einschlägt, hat sich das Räthsel gelöst: Carl Friedrich Gauß in Göttingen. CD-ROM. Elmar Mittler, Herausgeber. Göttingen: Staats- und Universitätsbibliothek. Available online. Link to Web page (Viewed 2006-02-02)

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.

Olson, Steve. 2004. Count Down: Six Kids Vie for Glory at the World's Toughest Math Competition. Boston: Houghton-Mifflin Company. (p. 67)

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.

Omnes, Roland. 1999. Quantum Philosophy: Understanding and Interpreting Contemporary Science. Princeton, N.J.: Princeton University Press. (p. 99)

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.

Pappas, Theoni. 1994. Fractals, Googols and other Mathematical Tales. San Carlos, Calif.: Wide World Publishing/Tetra.

Gauss shows off his math skill

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.

Park, David. 1999. Karl Friedrich Gauss finds a pattern. Mathematica notebook. Tutorial for high school students. Link to Mathematica notebook (Viewed 2006-02-15)

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 from 1 to 100. The students knew nothing about advanced arithmetic and had only one way of doing it. Let's see: 1 + 2 = 3, 3 + 3 = 6, 6 + 4 = 10, 10 + Oh, what was the last number we used? Well, you can see that it is a tedious calculation. And in those days there was no Mathematica. After all, it was only 1784. In those days all that the students had were slates which they could write on with chalk. When they had the answer, each would bring their slate up and lay it on the teacher's desk, one on top of the other. Büttner was full of glee with the misery he was causing the students.

But Büttner had no longer settled down than Karl Friedrich walked up, flung his slate onto the desk and said: "Ligget se." - "There it lies." He then sat down and folded his hands at his desk. It was a long time before another student brought up his slate, and then slowly, one by one, with a rush toward the end of the hour, the other students brought up their slates. Büttner looked at the top slate. The answer was wrong. The next one was wrong. Wrong, wrong, wrong! They were all wrong. Until he came to the bottom slate, Karl Friedrich's slate. It had the correct answer.

Do you know how Karl Friedrich Gauss was able to solve the problem so quickly - and get the correct answer? He did it by finding a pattern. Do you know what the pattern is? Once you see the pattern, the problem is easy. Without the pattern, it is futile drudgery. Finding patterns is the very essence of good mathematics.

We are going to use Mathematica to help see the pattern. It also gives us the opportunity to learn a few tricks of Mathematica.


[ ... ]
sum     1,   2,   3,   4 ...
sum   100,  99,  98,  97 ...
2 sum 101, 101, 101, 101 ...

Now we have a simple pattern. The numbers are all the same! We don't have to do 99 additions. We can do one multiplication. How many 101's are there? If you are in doubt, the top row just counts them. There are 100 of them. But notice that the bottom row is twice the sum because it contains each number twice. So our final answer is...

100*101/2 = 5050

So, "There it lies." That's the answer that Karl Friedrich wrote on his slate.

Pascual i Gainza, Pere. 2005. Geometria de superfícies: Una aproximació a la figura de Gauss. Lecture notes, Universitat Politècnica de Catalunya. Link to PDF file (Viewed 2006-02-15)

Els primers anys.

Hi ha moltes anècdotes referents a la precocitat matemàtica del jove Gauss, encara que els biògrafs assenyalen que bona part d'aquestes anècdotes es basen en els relats que feia el propi Gauss en els seus darrers anys de vida, cosa que dificulta la seva comprovació. Una de les més conegudes i contrastades és la del «descobriment» de la suma d'una progressió aritmètica: als nou anys Gauss va assistir a la seva primera classe d'aritmètica, on el professor Büttner va proposar als estudiants que calculessin la suma dels cent primers números. Gauss de seguida, va lliurar la seva pissarra al professor amb el resultat correcte, 5050, dient en el seu dialecte local, Ligget se!, és a dir, «aquí està». El jove Gauss havia advertit que la suma del primer i del darrer número donava el mateix resultat que la suma del segon idel penúltim, etc, és a dir,

1 + 100 = 2 + 99 = 3 + 98 =  ...  = 101,
i com que hi ha 50 parells, el resultat se segueix multiplicant: 101×50 = 5050.

Büttner va saber veure la capacitat de Gauss i el va posar a estudiar amb el seu ajudant Martin Bartels, de 17 anys, amb el qual Gauss va poder familiaritzar-se amb el binomide Newton per a exponents no enters, ambles sèries infinites i, va fer els primers passos en l'anàlisi matemàtica.

Pereira, Egmon. Undated Web site. Carl Friendrich Gauss. Link to Web page (Viewed 2006-02-02)

Em toda história da matemática, nunca houve uma criança tão precosse como Gauss - por sua própria iniciativa, trabalhou os rudimentos da aritmética antes de poder falar. Um dia, antes de ter completado três anos, seu gênio tornou - se aparente para seus pais de um modo muito contundente. Seu pai estava preparando a folha de pagamento semanal dos trabalhadores sob sua responsabilidade, enquanto o garoto observava calmamente de um canto. No fim dos cálculos, longos e cansativos Gauss disse ao seu pai que havia um erro no resultado e deu a resposta que ele obteve de sua cabeça. Para grande surpresa de seus pais, a verificação dos cálculos mostrou que Gauss estava certo!

Para sua educação elementar, Gauss foi matriculado em uma escola fraca dirigida por um homem chamado Büttner, cuja principal técnica de ensino era o espancamento. Büttner tinha por hábito passar longos problemas de adição que, desconhecido de seus alunos, eram progressões aritméticas que ele resolvia usando fórmulas. No primeiro dia em que Gauss entrou na aula de aritmética, foi pedido aos alunos que somassem os números de 1 a 100. Mas nem bem Büttner havia terminado de anunciar o problema, Gauss mostrou sua lousa e exclamou em seu dialeto camponês "ligget se" (Aqui jaz). Por quase uma hora Büttner fitou Gauss, que ficou sentado com os dedos entrelaçados enquanto os colegas esfalfavam. Quando Büttner examinou as lousas no final da aula, a lousa de Gauss continha um único número, 5050 - a única solução correta na classe. Para seu crédito, Büttner reconheceu o gênio de Gauss e com a ajuda de seu assistente, John Bartels, levou-o ao conhecimento de Karl Wilheln Ferdinand, duque de Brunswick.

Pérez Sanz, Antonio. Undated Web site. Carl Friedrich Gauss (1777-1855): "El príncipe de los matemáticos." Link to Web page (Viewed 2006-02-02)

En el seno de esta humilde familia, muy alejada de los salones ilustrados de la nobleza germana, el joven Gauss va a dar muestras tempranas de su genio precoz. Él mismo, ya anciano, acostumbraba a alardear de haber aprendido a contar antes que a escribir y de haber aprendido a leer por sí mismo, deletreando las letras de los nombres de los parientes y amigos de la familia. Y a él le debemos el relato de la anécdota que le coloca como el más precoz de los matemáticos. Cuando tenía tan sólo tres años, una mañana de un sábado de verano, cuando su padre procedía a efectuar las cuentas para abonar los salarios de los operarios a su cargo, el niño le sorprende afirmando que la suma está mal hecha y dando el resultado correcto. El repaso posterior de Gerhard dio la razón al niño. Nadie le había enseñado los números y mucho menos a sumar.

"Ligget se!" (¡Aquí está!)

A los siete años, tras serios esfuerzos de Dorothea para convencer al padre, Gauss ingresa en la escuela primaria, una vieja escuela, la Katherinen Volkschule, dirigida por J.G Büttner, donde compartirá aula con otros cien escolares. La disciplina férrea parecía ser el único argumento pedagógico de Büttner, y de casi todos los maestros de la época.

A los nueve años Gauss asiste a su primera clase de Aritmética. Büttner propone a su centenar de pupilos un problema terrible: calcular la suma de los cien primeros números. Nada más terminar de proponer el problema, el jovencito Gauss traza un número en su pizarrín y lo deposita en la mesa del maestro exclamando: "Ligget se!" (¡Ahí está!). Había escrito 5.050. La respuesta correcta.

Ante los ojos atónitos de Büttner y del resto de sus compañeros, Gauss había aplicado, por supuesto sin saberlo, el algoritmo de la suma de los términos de una progresión aritmética. Se había dado cuenta de que la suma de la primera y la última cifra daba el mismo resultado que la suma de la segunda y la penúltima, etc., es decir: 1+100 = 2+99 = 3+98 = ... = 101

Como hay 50 parejas de números de esta forma el resultado se obtendrá multiplicando 101 × 50 = 5.050

"Ligget se!"

Pettit, Annie. 2001. Carl Friedrich Gauss (1777-1855). Honors paper, Miami University of Ohio. Link to Web page (Viewed 2006-02-02)

As a child, Gauss was a prodigy. This event happened just before Gauss turned three years old.

"One Saturday Gerhard Gauss (his father) 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, Gerhard 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 Gauss was correct" (Bell 221).

What makes this more amazing is that nobody had taught him arithmetic. He picked it up on his own. Although Gauss showed great intelligence, his father refused to send him to school. His family was very poor as his father worked as a gardener, canal tender, and bricklayer (Bell 218). His dad wanted his son to follow in the family's footsteps and work as a laborer. However, his mother intervened and sent him to school when he was seven. His teacher, Büttner, was a cold-hearted teacher who loved proving to his students how ignorant they were (Bell 221). At the age of ten, Gauss "discovered" a formula that would change his future forever. Büttner asked his students to add up the numbers between one and a hundred. He figured this would keep his students busy all day. However, Gauss noticed a pattern. Without anyone showing him the formula [n(n+1)]/2, Gauss derived it and solved the problem quickly (Burton 510). Büttner was so impressed by this that he bought Gauss a math book and had his assistant, Johann Bartels, work with the young boy (Bell 222).

Planetmath.org. Undated Web site. Carl Friedrich Gauss. Link to Web page (Viewed 2005-11-25)

Carl Friedrich Gauss was born in Braunschweig on April the 30th 1777 as the son of a poor worker. Already in his youth he was interested in mathematics. It is reported that when Gauss was a student at elementary school his teacher asked the students to add up all natural numbers from 1 to 100, hoping to keep his students busy for some time. Gauss however found the correct answer within a few minutes by cleverly rearranging the summands.

Pólya, George. 1962. Mathematical Discovery: On Understanding, Learning, and Teaching Problem Solving. Vol. 1. New York: John Wiley & Sons. (pp. 60–61)

There is a traditional story about the little Gauss who later became the great mathematician Carl Friedrich Gauss. I particularly like the following version which I heard as a boy myself, and I do not care whether it is authentic or not.

"This happened when little Gauss still attended primary school. One day the teacher gave a stiff task: To add up the numbers 1, 2, 3, and so on, up to 20. The teacher expected to have some time for himself while the boys were busy doing that long sum. Therefore, he was disagreeably surprised as the little Gauss stepped forward when the others had scarcely started working, put his slate on the teacher's desk, and said, 'Here it is.' The teacher did not even look at little Gauss's slate, for he felt quite sure that the answer must be wrong, but decided to punish the boy severely for this piece of impudence. He waited until all the other boys had piled their slates on that of little Gauss, and then he pulled it out and looked at it. What was his surprise as he found on the slate just one number and it was the right one! What was the number and how did little Gauss find it?"

Of course, we do not know exactly how little Gauss did it and we shall never be able to know. Yet we are free to imagine something that looks reasonable. Little Gauss was, after all, just a child, although an exceptionally intelligent and precocious child. It came to him probably more naturally than to other children of his age to grasp the purpose of a question, to pay attention to the essential point. He just represented to himself more clearly and distinctly than the other youngsters what is required: to find the sum

     1
     2
     3
and so on
     .
     .
     .
    20
He must have seen the problem differently, more completely, than the others, perhaps with some variations as the successive diagrams A, B, C, D, and E. of Fig. 3.1 indicate. The original statement of the problem emphasizes the beginning of the series of numbers that should be added (A). Yet we could also emphasize the end (B) or, better still, the beginning and the end equally (C). Our attention may attach itself to the two extreme numbers, the very first and the very last, and we may observe some particular relation between them (D). Then the idea appears (E). Yes, numbers equally removed from the extremes add up all along to the same sum
1 + 20 = 2 + 19 = 3 + 18 = ... = 10 + 11 = 21
and, therefore, the grand total of the whole series is
10 × 21 = 210

Did little Gauss really do it this way? I am far from asserting that. I say only that it would be natural to solve the problem in some such way.

Posamentier, Alfred S., and Herbert A Hauptman. 2001. 101 Great Ideas for Introducing Key Concepts in Mathematics: A Resource for Secondary School Teachers. Thousand Oaks, Calif.: Corwin Press. (pp. 45–46)

As the story goes, young Gauss's teacher, Mr. Büttner, wanted to keep the class occupied, so he asked the class to add the numbers from 1 to 100. He had barely finished giving the assignment, when young Gauss put his slate down with simply one number on it, the correct answer! Of course, Mr. Büttner assumed Gauss had the wrong answer or cheated. In any case, he ignored this response and waited for the appropriate time to ask the students for their answers. No one, other than Gauss, had the right answer. What did Gauss do to get the answer mentally? Gauss explained his method:

Rather than to add the numbers in the order in which they appear,

1 + 2 + 3 + 4 + ... + 97 + 98 + 99 + 100
he felt it made more sense to add the first and the last, then add the second and the next-to-last, then the third and the third-from-the-last, and so on. This led to a much simpler addition:
1+100 = 101   2+99 = 101   3+98 = 101   4+97 = 101
Simply put, there are 50 pairs of numbers, each of which totals 101. Therefore the desired sum is 50 × 101 = 5050.

Although many textbooks may mention this cute story, they fail to use Gauss's technique to derive a formula for the sum of an arithmetic progression.

We generalize Gauss's method of addition. Consider the arithmetic progression sum

S = a + [a+d] + [a+2d] + [a+3d] + ... 
                       + [a+(n-2)d] + [a+(n-1)d]
where a is the first term, d is the common difference between terms, and n is the number of terms assumed at first to be even. Following Gauss's method of adding the first and nth terms gives:
a + [a + (n-1)d] = 2a + (n-1)d
Adding the second and (n-1)th term gives
 [a+d] + [a + (n-2)d] = 2a + (n-1)d
Adding the third and (n-2)th term gives
 [a+2d] + [a + (n-3)d] = 2a + (n-1)d
If we continue in this same manner until all the pairs have been added, we obtain n/2 such pairs, which gives the formula
S = (n/2)[2a + (n-1)d]
Students should now derive the same formula when n is odd....

[Footnote] According to E. T. Bell in his famous book, Men of Mathematics (Simon and Schuster, New York, 1937), Gauss, in his adult years, told this story, but explained that the situation was far more complicated than the simple one we currently tell. He told of his teacher, Mr. Büttner, giving the class a five-digit number, such as 81,297, asking them to add a three digit number, such as 198, to it 100 times successively, and then find the sum of that series. We can only speculate about which version is true!

Prasad, Ganesh. 1933. Some Great Mathematicians of the Nineteenth Century: Their Lives and their Works. In three volumes. Benares, India: The Benares Mathematical Society. (pp. 2–3)

Gauss received his earliest education at home and from 1784 to 1788 at a primary school of his native city. While a student at that school he met Johann Martin Christian Bartels (1769–1836), who was then an assistant teacher and later studied higher Mathematics and became Professor at the University of Dorpat. It is related that at this school during Gauss's 10th year an event took place which produced a great impression on the teachers and the students. It was the custom that soon after a sum was set by the teacher the students would begin to work it and the first boy to finish the work would place his slate on the table, the others following in succession as soon as they became ready with the solution. It soon happened that soon after his admission into the Arithmetic class, when a sum was set to the class Gauss put his slate within a minute of the announcement of the sum. After an hour many students finished the working and still it was found that Gauss's answer was right. It became then clear to the teacher, a man named Büttner, that Gauss had nothing to learn from him.

Rassias, George M. 1991. The Mathematical Heritage of C. F. Gauss. Singapore: World Scientific. (p. 1)

His father was a farmer who wished that his young son to follow one of the family trades and become a bricklayer or a gardener. But at a very early age it was clear that his son had unusual talents. He is said to have corrected an error to his father's payroll accounts at the age of three. At elementary school, at the age of eight, he added up all the numbers from one to a hundred in his first lesson. Recognizing his precocious talent, the teacher persuaded his father that Gauss should be encouraged to train to pursue a profession rather than learn a trade.

Rebolledo, Raquel Ardia de, Mario Ernesto Pérez Ruiz, Carmen Samper de Caicedo and Celly Serrano de Plazas. 2005. Espiral 11: Serie de matemáticas para secundaria y media. Bogatá: Grupo Editorial Norma. (p. 113)

Gauss fue un niño precoz. Se cuenta que a muy temprana edad, antes de entrar a la escuela, descubrió un error en la nómina semanal que su padre estaba elaborando para los trabajadores que tenía a su cargo y tambien que aprendió a leer por si mismo.

Se cuenta que en la escuela, cuando tenía 10 años, su profesor un tal Buttner, castigó a sus alumnos poniéndoles como tareo que sumaran los números de 1 hasta 100. Se acustumbraba que el alumno que terminara la tarea pusiera la pizarra sobre la mesa. Tan pronto el profesor acababa de anunciar la tarea, Gauss dijo "aqui está". Cuando Buttner le preguntó cómo lo había hecho, Gauss le dijo: 1+100 = 101, 2+99=101,... 3+98=101,..., siempre suman 101. Como son 50 sumas de 101, el resultado es 5050.

Reich, Karin. 1977. Carl Friedrich Gauss: 1777–1977. Translated by Patricia Crampton. Bonn-Bad Godesberg: Inter Nationes. (pp. 7–8)

In 1784, at the age of 7, Gauss went to primary school, the Katharinenschule in Brunswick, headed by the teacher J. G. Büttner. In conformity with conditions in those days Büttner was teaching about a hundred children in one class, and chastisements, carried out with the "Karwatsche" (whip) were a matter of course. For two years there seemed to be nothing unusual about the schoolboy Gauss. This changed when in the third year of school he entered the mathematics class. The custom then was that when sums were being done during the school period, the pupil to finish first put his slate on the teacher's table, the second quickest laid his slate on top of the first, etc. At the very beginning of the school year, in order to keep his pupils occupied for some time, Büttner set the task of adding together the numbers from 1 to 100, i.e., 1 + 2 + 3 + ... + 98 + 99 + 100. Scarcely had the teacher explained the task when Gauss laid his slate on the desk and announced in the Brunswick dialect: "Ligget se!" ("There it is!"). On the slate was written a single figure, 5050. While the rest of the pupils—just as today's pupils would at that age—were beginning to count up 1 + 2 = 3, 3 + 3 = 6, 6 + 4 = 10, ..., it occurred to Gauss that the first and last figures together produced the same result as the second and last-but-one, etc., that is 1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101 ... with this assumption the whole task was reduced to multiplying 101 by 50, which can easily be done in one's head. Büttner soon recognized the extraordinary mathematical talent of his pupil and admitted that Gauss had learned all he could in his school.

Ross, Peter. 1995. Why isn't there a Nobel prize in mathematics? Math Horizons, November 1995, p. 9. Available online. Link to Web page (Viewed 2005-11-24)

There is a larger question raised by the fact that apocryphal stories, such as the Nobel-math-prize myth, seem to have a life of their own.... Another example of this tendency concerns the famous story of Gauss's discovery as a ten-year old boy of a simple method for summing an arithmetic series. (Multiply the number of terms by the average of the smallest and largest terms.) Most mathematicians who teach will assert that the problem given to Gauss by his tyrannical school teacher was to sum the integers from 1 to 100. In fact, Gauss was given a more difficult problem "of the following sort, 81297 + 81495 + 81693 +... + l00899, 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." (p. 221 of E. T. Bell's Men of Mathematics, 1937). With this particular example it's easy to maintain historical truth by telling students that Gauss was given a problem like summing the integers from 1 to 100.

Ryan, Jordan and Matthew. 2001. Math Mania: Johann Carl Friedrich Gauss. Flint Hill Elementary School, Vienna, Va. Link to Web page (Viewed 2006-02-02)

Johann Carl Friedrich Gauss was born on April 30, 1777. He was born in Brunswick, Duchy of Brunswick, which is now Germany. He died on February 23, 1855 in Göttingen, Hanover, which is now Germany. Gauss was always fascinated by mathematical ideas. It has been said that at the age of 3 Gauss corrected his father's computations.

When Gauss was in elementary school his teacher Master Büttner did not really like math so he did not spend a lot of time on the subject. One of the problems his teacher gave the class was "add all the whole numbers from 1 to 100". His teacher Master Büttner was amazed that Gauss could add all the whole numbers 1 to 100 in his head. Master Büttner didn't believe Gauss could do it, so he made him show the class how he did it. Gauss showed Master Büttner how to do it and Master Büttner was amazed at what Gauss just did. The system of how he did it is add 1+100, 2+99, 3+98 ... 49+52 and he had 50 pairs of 101 and he multiplied 101×50 to get 5050, which is the answer.

Sartorius von Waltershausen, W. 1856. Gauss: zum Gedächtniss. Leipzig: Verlag von S. Hirzel. (pp. 10–13)

Gauss bewahrte dem engen kleinen Kreise des elterlichen Hauses, worin seine erste Jugend verstrich, bis an sein Lebensende ein Andenken voll rührender Pietät und wandte gern noch im hohen Alter seine Erinnerung auf unzählige kleine charakteristische Züge aus seiner frühsten Kindheit zurück, welche die äuserlich beschränkten, bescheidenen Verhältnisse derselben wiederspiegeln und in denen man die wunderbare Begabung des später so grossartig entfalteten Geistes schon einzelne Funken sprühen sieht. Er hatte sie treu im Gedächtniss behalten und wusste durch seine heiter gemültliche, lebendige Erzählungsweise, worin bei ihrer Wiederholnung nie die kleinste Abweichung vorkam, einen erhöhten, unbeschreiblich lieblichen Reiz ihnen zu verleihen, der im todten Buchstaben, wenn wir versuchen wollten einzelne davon hier wiederzugeben, leider verloren gehen würde. &emdash; In seine frühste Jugendzeit reichte seine Erinnerung daran zurück, das er als kleines Kind einst nahe dem Tode gewesen war. Der vorerwähnte Wendengraben, an welchem seine Eltern wohnten, gegenwärtig übermauert, war früher ein offener mit der Ocker in Verbindung stehender Canal, im Frühling mit Wasser reichlich erfüllt. Der kleine unbeaufsichtigt daran spielende Knabe fiel hinein und wurde, eben vor dem Ertrinkin, wie durch die Hand der Vorsehung gerettet, um für die höchsten wissenschaftlichen Leistungen, zum Ruhme unseresVaterlands aufbewahrt zu werden.

Schon in seinen ersten Lebensjahren gab Gauss die ausserordentlichsten Beweise seiner geistigen Fähigkeiten. Nachdem er den Einen und den Andern der Hausbewohner um die Aussprache der Buchstaben gebeten hatte, erlernte er das Lesen von selbst, soch ehe er die Schule besuchte, und zeigte einen so bewunderungswürdigen Sinn für die Auffassung von Zahlenverhältnissen und eine so unglaubliche Leichtigkeit und Sicherheit im Kopfrechnen, dass er dadurch sehr bald die Aufmerksamkeit seiner Eltern und die Theilnahme nahestehender Freunde erregt hat. Er selbst pflegte oft scherzweise zu sagen, er habe früher rechnen als sprechen können.

Gauss' Vater betrieb den Sommer über ein Maurer-Handwerk. Am Sonnabend pflegte er für die geschlossene Woche seinen unter ihm arbeitenden Gesellen den Lohn auszuzahlen, bei welcher Gelegenheit jenen, welche nach dem Feierabend gearbeitet hatten, für jede einzelne Stunde ihrer ausserordentlichen Beschäftigung eine dem Tagelohn verhältnissmässige Vergütung zugeschrieben wurde. Nachdem der Meister für die verschiedenen Betheiligten seine Rechnung geschlossen hatte, und im Begriff war das Geld zu verabfolgen, erhebt sich der kaum dreijährige Knabe, der unbemerkt den Verhandlungen seines Vaters gefolgt war, von seinem ärmlichen Lager und ruft mit kindlicher Stimme: "Vater, die Rechnung ist falsch, es macht so viel," indem er eine gewisse Zahl nannte. Die Rechnung wurde darauf mit grosser Aufmerksamkeit wiederholt und zum Erstaunen aller Anwesenden genau so gefunden, wie sie von dem Kleinen angegeben war.

Gauss besuchte suerst 1784, nachdem er sein siebentes Lebensjahr zurückgelegt, die Catharinen-Volksschule, in welcher der erste Elementar-Unterricht ertheilt wurde und die damals unter der Leitung eines gewissen Büttner gestanden hat. Es war eine dumpfe, niedrige Schulstube mit einem unebenen ausgelaufenen Fussboden, von der man nach der einen Seite gegen die beiden schlanken gothischen Thürme der Catherinen-Kirche, nach der andern gegen Ställe und armselige Hintergebäude hinaus blickte. Hier ging Büttner zwischen etwa hundert Schülern auf und ab, mit der Karwatsche in der Hand, welche damals als ultima ratio seiner Erziehungsmethode von Gross und von Klein anerkannt wurde und von der er nach Laune und Bedürfniss einen schonungslosen Gebrauch zu machen sich berechtigt fühlte. In dieser Schule, die noch sehr den Zuschnitt des Mittelalters gehabt zu haben scheint, blieb der junge Gauss zwei Jahre lang ohne durch etwas Ausserordentliches aufzufallen. Erst nach jener Zeit brachte es der Gang des Unterrichtes mit sich, dass auch er in die Rechenklasse eintrat, in welcher die Meisten bis zu ihrer Confirmation, bis etwa zu ihrem 15ten Jahre blieben. Es ereignete sich hier ein Umstand, den wir nicht ganz unbeachtet lassen dürfen, da er auf Gauss' späteres Leben von einigem Einfluss gewesen ist und den er uns in seinem hohen Alter mit grosser Freude und Lebhaftigkeit öfter erzählt hat. Das Herkommen brachte es nämlich mit sich, dass der Schüler, welcher zuerst sein Rechenexempel beendigt hatte, die Tafel in die Mitte eines grossen Tisches legte; über diese legte der zweite seine Tafel u.s.w. Der junge Gauss war kaum in die Rechenclasse eingetreten, als Büttner die Summation eine arithmetischen Reihe aufgab. Die Aufgabe war indess kaum ausgesprochen als Gauss die Tafel mit den im niedern Braunschweiger Dialekt gesprochenen Worten auf den Tisch wirft: "Ligget se'". (Da liegt sie.) Während die andern Schüler emsig weiter rechnen, multipliciren und addiren, geht Büttner sich sich seiner Würde bewusst auf und ab, indem er nur von Zeit zu Zeit einem mitleidigen und sarcastischen Blick auf den kleinsten der Schüler wirft, der längst seine Aufgabe beendigt hatte. Dieser sass dagegen ruhig, schon eben so sehr von dem festen unerschütterlichen Bewusstsein durchdrungen, welches ihn bis zum Ende seiner Tage bei jeder vollendeten Arbeit erfüllte, dass seine Aufgabe richtig gelöst sei, und dass das Rusultat kein anderes sein könne. Am Ende der Stunde wurden darauf die Rechentafeln ungekehrt; die von Gauss mit einer einzigen Zahl lag oben und als Büttner das Exempel prüfte, wurde das seinige zum Staunen aller Anwesenden als richtig befunden, während viele der übrigen falsch waren und alsbald mit der Karwatsche rectificirt wurden. Büttner glaubte nun ein gutes Werk zu thun eigens aud Hamburg ein neues Rechenbuch zu verschreiben, um damit den jungen bahnbrechenden Geist nach Kräften zu unterstützen, er soll aber einsichtsvoll genug gewesen sein bald zu erklären, dass Gauss in seiner Schule nichts mehr lernen könne.

Sartorius von Waltershausen, W. 1856, 1966. Carl Friedrich Gauss: A Memorial. Translated by Helen Worthington Gauss. 1966. Colorado Springs, Colo. (pp. 2–4)

As long as he lived Gauss cherished memories of the narrow little homecircle of his childhood. In old age he liked to recall characteristic little episodes which reflected the outwardly restricted, modest homelife, but in which one detected the sparks of genius which later rose to such heights. He remembered these incidents accurately and in recounting them he never varied the details. His amused and cheerful telling gave them a delightful charm which would be lost were they to be repeated in print.

When very small, he recalled, he was once near death. The Wendengraben on which his parents lived was in those days an open canal connected with the Ocker river, and in the spring was full of water. The little fellow was playing by it one day, when he fell in, to be saved just as he was sinking, as if preserved by Providence for his high achievements in the world of Science.

While still very young Gauss showed rare mental gifts. He learned to read by asking one or another in the home the sound of the letters. His marked aptitude for numbers and his ease and accuracy in mental arithmetic soon attracted the attention of his parents and their friends. He used to say jestingly that he learned to count before he could talk.

Gauss's father carried on in the summer a masonry business. On Saturday evenings it was his habit to pay his workmen their past week's wages, paying those who had worked overtime according to the extra hours they had put in. On one such occasion he had finished the reckoning and was about to pay out the money when there came a childish voice from a small bed in the corner of the room. Unnoticed the three-year old child had been following his father's transactions. Now he said, "Father, the reckoning is wrong. It is so much," naming a certain figure. The reckoning was gone over again and was found to be what the child had said.

In 1784 after his seventh birthday the little fellow entered the public school where elmentary subjects were taught and which was then under a man named Büttner. It was a drab, low school-room with a worn, uneven floor. On one side one looked out on the two slender Gothic towers of the Catharinen Church, on the other side were stables and poor back-yard dwellings. Here among some hundred pupils Büttner went back and forth, in his hand the switch which was then accepted by everyone as the final argument of the teacher. As occasion warranted he used it. In this school—which seems to have followed very much the pattern of the Middle Ages—the young Gauss remained two years without special incident. By that time he had reached the arithmetic class in which most boys remained up to their fifteenth year.

Here occurred an incident which he often related in old age with amusement and relish. In this class the pupil who first finished his example in arithmetic was to place his slate in the middle of a large table. On top of this the second placed his slate and so on. The young Gauss had just entered the class when Büttner gave out for a problem the adding of a series of numbers from 1 to 100. The problem was barely stated before Gauss threw his slate on the table with the words (in the low Braunschweig dialect): "There it lies." While the other pupils continued busily adding, Büttner, with conscious dignity, walked back and forth, occasionally throwing an ironical, pitying glance toward this the youngest of the pupils. The boy sat quietly with his task ended, as fully aware as he always was on finishing a task that the problem had been correctly solved and that there could be no other result.

At the end of the hour the slates were turned bottom up. That of the young Gauss with one solitary figure lay on top. When Büttner read out the answer, to the surprise of all present that of young Gauss was found to be correct, whereas many of the others were wrong. Büttner now decided to write to Hamburg for a new book on arithmetic which would be better suited to the young lad's exceptional mind. But before long he is said to have had enough insight to declare that Gauss could learn nothing more in his school.

Schaaf, William L. 1964. Carl Friedrich Gauss: Prince of Mathematicians. Immortals of Science series. New York: Franklin Watts. (pp. 5–7)

When Carl was seven years old, he was sent to a local Volksschule, or high school, which was presided over by a tyrannical teacher who made free use of a whip to control the boys. Some three years later, Gauss entered the arithmetic class. Here there occurred an episode that doubtless influenced his later life, and which, in any event, has become a classic anecdote. One day, to keep the boys busy, the teacher gave the class the exercise of writing down all the numbers from 1 to 100 and then finding their sum. This exercise was designed to keep them occupied for quite a while. (Try it sometime!) The pupil who finished the task first was told to put his slate on a big table; the second one to finish would then place his slate on top of this, and so forth. Hardly had the teacher given the boys this problem, when Gauss threw his slate upon the table, saying: "Ligget se!" which in the low German peasant dialect of the time meant "There it is!" While the other pupils were busily writing down the numbers and adding them up, the pompous teacher marched up and down, frowning scornfully at the youthful Gauss. Finally, upon looking at Gauss's slate, he found but one number written there. It was the correct answer—5,050.

How had Gauss done it? The answer is that he had succeeded in recognizing an obvious pattern:

   1   2   3   4  .......  97   98   99   100
(1+100)=101; (2+99)=101; (3+98)=101; etc.
Clearly there are 50 such pairs, all having the same sum, 101; therefore the sum of all the numbers is 50×101, or 5,050.

Why not try your own hand at this kind of thinking?

Schlesinger, Ludwig. 1933. Über Gauss' Arbeiten zur Funktionentheorie II. Traditionen über die erste Jugendzeit bis zum Abgang auf die Universität Oktober 1795. In Gauss Werke, herausgegeben von der K. Gesellschaft der wissenschaften zu Göttingen. Göttingen: Gedruckt in der Dieterichschen universitäts druckerei (W. F. Kaestner). Vol. 10, Part 2. Available online from Göttinger Digitalisierungs-Zentrum. Link to PDF file

Gauss' Mutter, die als eine feinsinnige Frau von festem und heiterem Charakter geschildert wird, ist 1817 als 74 jährige Witwe zu ihrem Sohne nach Göttingen gezogen, wo sie noch 22 Jahre be ihm auf der Sternwarte gewohnt hat. Ihren Erzählungen verdankt man Nachtrichten über geistige Entwicklung von Gauss in den ersten Jahren seiner Kindheit.

Schon kaum dreijährig habe er den Vater bei der Lohnauszahlung am Samstag Abend auf einen Rechenfehler aufmerksam gemacht, das Lesen habe er sich selbst ohne Unterricht angeeignet. Aus seinem siebenten Lebensjahr erzählt Gauss selbst, er habe in der Schule eine langwierige Rechenaufgabe in wenigen Augenblicken gelöst, indem er auf das Gesetz der arithmetischen Reihe aufmerksam wurde, und während seine Kameraden sich eben anschickten, durch mühsames Addieren die Lösung zu finden, habe er seine nur eine einzige Zahl enthaltende Tafel mit dem übermütigen Ausruf im Braunschweiger Dialekt: "Dar licht se" auf das Katheder geworfen. Der Eindruck, den diese Leistung auf den Schulmeister Büttner hervorgerufen hat, bewirkte, dass dieser für den ungewöhnlichen Schüler aus Hamburg ein Rechenbuch kommen liess, Remers Arithmetica, das sich mit der Eintragung: "Johann Friedrich Carl Gauss, Braunschweig, 16 December Anno 1785" noch in der Gauss-bibliothek befindet und ebenso wie das Exemplar von Hemelings Arithmetischen kleinen Rechenbuch Spuren starker Benutzung und zwischen dem Text einige von Gauss' kindlicher Hand ausgeführte elementare Rechnungen zeigt.

Scolnik, Hugo. 2006. Mi vida junto a Gauss. Link to Web page (Viewed 2006-02-03)

Antes de contar algunas de las muchas veces en que encontré con él, vale la pena recordar algunos hechos de su vida. Nació el 30 de Abril de 1777 en una casita miserable en Brunswick, y murió el 23 de febrero de 1855 en Göttingen, ambas ciudades alemanas. Hijo de un modesto albañil, desde los tres años dio muestras de su genio al aprender a leer y hacer cálculos mentalmente (le gustaba decir que había aprendido primero a contar y luego a hablar) Su genio pudo desarrollarse gracias a su madre Dorothea, quien hizo lo imposible para que su hijo no siguiese el camino que su padre quería imponerle.

Cuando tenía diez años ingresó a una escuela primaria de corte medieval donde un maestro llamado Büttner, quien aterrorizaba a todo el mundo, les pidió a los alumnos que sumaran todos los números de 1 a 100, con la sana idea de que no molestaran por un buen rato. Pero Gauss levantó la mano inmediatamente y dio la respuesta correcta: 5050. ¿Cómo lo había hecho? Pues si sumamos el primer y el último número resulta que 1 + 100 = 101, el segundo y el anteúltimo dan 2 + 99 = 101, etc. Si hay 100 números entonces hay 50 pares que suman 101, de donde resulta que 50×101 = 5050 era la solución. Había redescubierto las progresiones aritméticas.

Scripture, E. W. 1891. Arithmetical prodigies. The American Journal of Psychology 4(1):1–59. Available online. Link to Web page (p. 9)

An anecdote of his early life, told by himself, is as follows: His father was accustomed to pay his workmen at the end of the week, and to add on the pay for overtime, which was reckoned by the hour at a price in proportion to the daily wages. After the master had finished his calculations and was about to pay out the money, the boy, scarce three years old, who had followed unnoticed the acts of his father, raised himself and called out in his childish voice: "Father, the reckoning is wrong, it makes so much," naming a certain number. The calculation was repeated with great attention, and to the astonishment of all it was found to be exactly as the little fellow had said.

At the age of nine Gauss entered the reckoning class of the town school. The teacher gave out an arithmetical series to be added. The words were scarcely spoken when Gauss threw his slate on the table, as was the custom, exclaiming, "There it lies!" The other scholars continue their figuring while the master throws a pitying look on the youngest of the scholars. At the end of the hour the slates were examined; Gauss's had only one number on it, the correct result alone. At the age of ten he was ready to enter upon higher analysis. At fourteen he had become acquainted with the works of Euler and Lagrange, and had grasped the spirit and methods of Newton's Principia.

Simmons, John. 2000. The Scientific 100: A Ranking of the Most Influential Scientists, Past and Present. New York: Citadel Press. (p. 195)

A true mathematical prodigy, Gauss could do sums by age three, when he began to correct his father's addition. Sent to a provincial school at age seven, he began his arithmetic class two years later. The story is told that the schoolmaster gave the class some make-work: to add the first hundred integers. Gauss immediately grasped the principle of an arithmetic progression, wrote down the answer, and as the instructor finished with the sums, tossed down his slate, saying, Ligget se ("There it lies!").

Simon, Marvin K. 2002. Probability Distributions Involving Gaussian Random Variables: A Handbook for Engineers and Scientists. Norwell, Mass.: Kluwer Academic Publishers. (p. xiii)

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.

Stewart, Ian. 1977. Gauss. Scientific American 237(1):122–131.

Among the great mathematicians there are about as many who showed mathematical talents in chidhood as there are those who showed none at all until they were older. Gauss was unquestionably the most precocious of them all. He joked that he had been able to count before he could talk, and many anecdotes attest to his extraordinary gifts. It is said that one day before he was three years old his father was making out the weekly payroll for the laborers in his charge, unaware that his son was observing the process intently. Finishing his calculations, the elder Gauss was startled to hear a tiny voice saying: "Father, the reckoning is wrong, it should be...." When the computation was checked, the child's figure was found to be the right one. The remarkable thing about the story is that no one had taught him arithmetic.

Other tales concern Gauss's continued precocity at school. When he was 10, he was admitted to the arithmetic class. The master handed out a problem of the following type: What is the sum of 1 + 2 + 3 + ... + 100, where there are 100 numbers and the difference between each number and the next is always one. There is a simple trick for doing such sums that was known to the master but not to his pupils.

The custom was for the first boy who solved a problem to lay his slate on the master's table with the answer written on it, for the next boy to lay his slate on top of that one and so on. The master had barely finished stating the problem when Gauss put his slate on the table. "There it lies," he said. For the next hour he sat with folded arms, getting an occasional skeptical look from the master, as the other boys toiled at the addition. At the end of the hour the master examined the slates. On Gauss's was a single number. Even in his old age Gauss loved to tell how of all the answers his was the only correct one.

Suzuki, Jeff. 2002. A History of Mathematics. Upper Saddle River, N.J.: Prentice Hall. (p. 577)

Stories of Gauss's early mathematical genius abound: when he was 3, he uncovered an error in his father's bookkeeping; when Gauss was 8, Büttner, the local school teacher, gave to the class the problem of summing the first 100 whole numbers, to which Gauss gave the correct answer—5050—almost immediately and without any obvious computation. The teacher was impressed. The mythology of mathematics says the teacher was furious, but Büttner supplied Gauss with more advanced works and encouraged him to work with one of the assistant teachers, Johann Martin Bartels.

Swetz, Frank J. 1994. Learning Activities from the History of Mathematics. Portland, Maine: Walch Publishing. (p. 8)

The injection of one or more brief historical anecdotes during a class discussion offers a change of pace from the usual teacher-student interaction. It can be amusing, informative, and thought-provoking and set the stage for further mathematical discussion and learning. As an example, consider the story involving the young Gauss, the so-called Prince of Mathematicians, who at age ten was given the task by a teacher of finding the sums of the natural numbers from 1 to 100. His teacher thought this problem would keep the class occupied for some time (busywork is not new) and was amazed when Gauss quickly made a mental calculation and supplied the correct answer, 5,050. The young boy had noted this pattern:

  1 +   2 +   3 + ... +  98  + 99 + 100
100 +  99 +  98 + ... +   3  +  2 +   1
101 + 101 + 101 + ... + 101 + 101 + 101 = 100(101)
Realizing he had obtained twice the required sum, Gauss modified his answer by dividing by 2:
(100)(101) / 2 = 5,050

This simple story emphasizes the importance of looking at mathematics: seeking patterns and, if possible, using those patterns to solve a problem. While a powerful lesson in problem-solving has been carried out, further considerations and questions easily evolve: "Does the technique work for any consecutive series of numbers?" "Would it work for a series of even numbers?" "Can the class generalize a formula to encompass the process?" and so on.

Tent, M. B. W. 2006. The Prince of Mathematics: Carl Friedrich Gauss. Wellesley, Mass.: A K Peters.

After the first two years of school, it was time to begin arithmetic. One day when Herr Büttner wanted to keep the boys quiet for awhile, he gave them an assignment he knew would accomplish that. He asked them to find the sum of the first 100 numbers: 1 + 2 + 3 + 4 + 5 + 6 + 7 + ... + 96 + 97 + 98 + 99 + 100. The other boys quickly set to work on their enormous chore, adding 1 + 2 = 3; 3 + 3 = 6; 6 + 4 = 10; 10 + 5 = 15; 15 + 6 = 21, .... This was going to take them a long time, but Herr Büttner's whip was ready to straighten out any boy who gave up on the job.

Carl used a different approach. Rather than start in on the adding immediately, he sat and thought a minute. Then he wrote the answer on his slate and walked up to Herr Büttner's desk to put his slate on the desk as the first one in the pile. As the other boys finished, they would put their slates on top of Carl's in the order in which they finished. Herr Büttner looked at Carl's slate, saw just one number, and glared at Carl. What a pleasure it would be to correct that child when it was time to check the answers! In the meantime, Carl sat at his desk and waited patiently for the others to finish. As he did in later life, Carl knew that his solution was correct. If he had been easily intimidated, Herr Büttner's glances would have made him tremble. Gauss' confidence in his own reasoning then and in later years was unshakable. It was time to move on to the next challenge.

[ ... ]

An hour later, when all the slates were finally stacked and ready to be checked, Büttner reached into his pocket to get the slip of paper with the answer on it, and then he began to check the slates. One after another was found to have the wrong answer. Some boys had made their first mistake early in their calculations, so naturally they were doomed to failure almost from the beginning, and most of their hard work was completely wasted. Some waited until close to the end to make their first mistakes, but even so their answers were wrong. This assignment was proving to be a trial for most of the boys. The whip got plenty of use.

Finally Herr Büttner reached Carl's slate at the bottom of the pile, and there he found the correct answer: 5,050. How had Carl gotten it? He had spent almost no time on it, he hadn't done any figuring on his slate, and it looked suspicious. He demanded an explanation. "Tell me, boy, how you got this answer!" he demanded.

Carl stood up and began to speak. "Well, sir, I thought about it. I realized that those numbers were all in a row, that they were consecutive, so I figured there must be some pattern. So I added the first number and the last number: 1 + 100 = 101. Then I added the second and the next to last numbers: 2 + 99 = 101. It started to make sense. 3 + 98 = 101; 4 + 97 = 101; 5 + 96 = 101. If I continued adding pairs of numbers like that, I would eventually reach 50 + 51. That meant I would find fifty pairs of numbers that always add up to 101, so the whole sum must be 50 × 101 = 5,050. There was no need to add up all the numbers, sir."

Herr Büttner was dumbfounded. He had learned a formula to figure out problems like this. After all, he certainly hadn't sat down and added up all those numbers! But how could this ten-year-old boy, son of a common working man and illiterate mother, figure it out? It had taken Herr Büttner many years of schooling to master techniques like that, and now young Gauss had figured it out for himself in just a few minutes! Perhaps he had underestimated the boy.

Ullrich, Peter. 2005. Herkunft, Schul- und Studienzeit von Carl Friedrich Gauß. In Carl Friedrich Gauß in Göttingen, edited by Elmar Mittler. Göttinger Bibliotheksschriften 30. Göttingen: Niedersächsische Staats- und Universitätsbibliothek. Link to PDF file (Viewed 2006-03-11)

Über diese grundlegenden intellektuellen Fähigkeiten hinaus finden sich in den Gauß'schen Kindheitserzählungen bereits Hinweise auf seinen kreativen Umgang mit Zahlen, so in einem Bericht, der schildert, wie seiner Klasse im dritten Schuljahr die Aufgabe gestellt wurde, eine arithmetische Folge zu addieren. Gauß verließ sich dabei nicht auf seine Fähigkeiten im raschen nummerischen Rechnen – auch wenn diese ihm sicherlich gestattet hätten, die Addition schneller als alle seine Klassenkameraden durchzuführen – sondern benutzte, wenn man dem Bericht glauben darf, offenbar eine geschickte Vereinfachung der Rechnung: Zumeist heißt es, dass es um die Addition aller Zahlen von 1 bis 100 gegangen sei. Dabei lassen sich die erste und die letzte Zahl, also 1 und 100, zu einem Paar zusammenfassen, das die Summe 101 ergibt, ebenso die zweite und die vorletzte Zahl, 2 und 99, usw. Insgesamt erhält man so 50 Zahlenpaare, die jeweils die Summe 101 ergeben, mithin insgesamt 5050 als Gesamtresultat. Diese Lösung ist zwar nicht erst von Gauß entwickelt worden: Sie war schon in der griechischen Antike bekannt, etwa im 2. Jahrhundert v. Chr. bei Hypsikles, und findet sich auch in den um 800 n. Chr. entstandenen und Alkuin von York (um 735–804) zugeschriebenen “Propositiones ad acuendos iuvenes" [“Aufgaben zur Schärfung des Geistes der Jünglinge"]. Die Leistung des ungefähr neunjährigen Gauß lässt aber schon seine Fähigkeit erahnen, auch rein nummerische Rechnungen nicht schematisch, sondern unter Verwendung möglichst vieler Vereinfachungen und auch Kontrollen durchzuf?hren.

Vakil, Ravi. 1996. A Mathematical Mosaic: Patterns and Problem Solving. Burlington, Ontario: Brendan Kelly Publishing. (p. 137)

Gauss enjoyed numerical calculation as a child. An anecdote told of his early schooling reveals his precocious computing ability and his mathematical insight. One day, in 1787, when Gauss was only 10 years old, his 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 Gauss placed his slate on the table, saying, "Ligget se" ("There it lies"). The teacher looked at him scornfully while the others worked diligently. When the instructor finally looked at the results, the slate of Guass 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 by grouping the numbers in pairs to total 101 as shown below.

   1     2     3    4    5       48    49    50
+100   +99   +98  +97  +96  ... +53   +52   +51
 101   101   101  101  101      101   101   101
There are 50 pairs, each summing to 101, for a total sum of 50 × 101, or 5050.

Weisstein, Eric W. 2003. CRC Concise Encyclopedia of Mathematics. Second edition. Boca Raton, Fla.: CRC Press. (Entry on arithmetic series, pp. 123–124)

Note, however, that

a1 + an = a1 + [a1 + d(n-1)] = 2a1 + d(n-1),
so
Sn = 1/2 n (a1 + an),
or n times the average of the first and last terms! This is the trick Gauss used as a schoolboy to solve the problem of summing the integers from 1 to 100 given as busy-work by his teacher. While his classmates toiled away doing the addition longhand, Gauss wrote a single number, the correct answer
1/2 (100(1 + 100)) = 50 × 101 = 5050
on his slate (Burton 1989, pp. 80–81; Hoffman 1998, p. 207). When the answers were examined, Gauss's proved to be the only correct one.

Weisstein, Eric. Undated. Gauss, Karl Friedrich (1777-1855). Eric Weisstein's World of Biography. Link to Web page (Viewed 2005-11-25)

German mathematician who is sometimes called the "prince of mathematics." He was a prodigious child, at the age of three informing his father of an arithmetical error in a complicated payroll calculation and stating the correct answer. In school, when his teacher gave the problem of summing the integers from 1 to 100 (an arithmetic series) to his students to keep them busy, Gauss immediately wrote down the correct answer 5050 on his slate.

There is also a story that in 1807 he was interrupted in the middle of a problem and told that his wife was dying. He is purported to have said, "Tell her to wait a moment 'til I'm through" (Asimov 1972, p. 280).

Wikipedia article on Gauss. Undated. Carl Friedrich Gauss. Link to Web page (Viewed 2005-11-25)

Gauss was born in Brunswick (German: Braunschweig), in the Duchy of Brunswick-Lüneburg (now part of Lower Saxony, Germany), as the only son of uneducated lower-class parents. According to legend, his gifts became apparent at the age of three when he corrected, in his head, an error his father had made on paper while calculating finances. Another story has it that in elementary school his teacher tried to occupy pupils by making them add up the integers from 1 to 100. The young Gauss produced the correct answer within seconds by a flash of mathematical insight, to the astonishment of all. Gauss had realized that pairwise addition of terms from opposite ends of the list yielded identical intermediate sums: 1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101, and so on, for a total sum of 50 × 101 = 5050 (see arithmetic series and summation). While the story is mostly true, the problem assigned by Gauss's teacher was actually a more difficult one.

Wikipedia.de Artikel "Der kleine Gauss". Link to Web page (Viewed 2006-02-04)

Der kleine Gauß bezeichnet eine mathematische Formel, die von dem Mathematiker Carl Friedrich Gauß als kleiner Junge angewandt wurde. Sie war allerdings schon seit Jahrhunderten bekannt.

Mit neun Jahren wurde Gauß in der Schule Martino-Katharineum die Aufgabe gestellt, die Zahlen von 1 bis 100 zu summieren. Zur Verwunderung seines Lehrers hatte er sie nach kurzer Zeit gelöst. Er addierte 50 mal die Zahl 101, die sich aus der Summe von jeweils zwei Paaren ergibt (1+100, 2+99, ..., 50+51). Als Ergebnis erhielt er richtigerweise 5050.

Die allgemeine Formel für die Summe s von allen natürlichen Zahlen zwischen 0 und n lautet nun

 s = Σ k = n(n+1)/2
Anschaulich

Anschaulich machen kann man sich die Berechnung, indem man sich die 100 Zahlen als Reihen von Knödeln in 100 Zeilen vorstellt, ein Knödel in der ersten Zeile, der zweite Knödel in der zweiten Zeile usw. Wenn man dieses Dreieck jetzt um 180 Grad dreht und nocheinmal daneben stellt, erhält man 100 Zeilen, die jede genau 101 Knödel enthalten. Die Knödelanzahl dieses Rechtecks ist leicht zu berechnen, sie beträgt 100 mal 101 Knödel. Das zuvor hinzugefügte Dreieck wieder weggenommen, bleibt genau die Hälfte der Knödel übrig.

So sähe das für die Zahlen eins bis fünf aus:

o ooooo
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Eine weitere Möglichkeit, die Berechnung zu veranschaulichen ist die Folgende:

Möchte man beispielsweise die Zahlen von 1 bis 10 aufsummieren, so schreibe man die Zahlen nebeneinander in eine Zeile. In die Zeile darunter notiere man die Zahlen in umgekehrter Reihenfolge:

 1  2  3  4  5  6  7  8  9  10
10  9  8  7  6  5  4  3  2   1
Es ist gut zu erkennen, dass die Summe der Spalten jeweils den Wert 11 ergibt.

Da man dieses Ergebnis bei allen 10 Spalten erhält ergibt sich für die Summe aller Ziffern im Kasten der Wert 10 × 11. Möchte man nun den Wert der Summe einer Zeile wissen, so ist das Ergebnis zu halbieren und es ergibt sich (10 × 11)/2.

Worbs, Erich. 1955. Carl Friedrich Gauss: Ein Lebensbild. Leipzig: Koehler & Amelang. (pp. 17–18)

Bezeichnend ist ein Erlebnis, das der Lehrer Büttner der Braunschweiger Anfängerschule mit ihm hat. Noch ist es die Zeit vor der Neugestaltung des Braunschweiger Schulwesens, und ein mechanischer Drill ohne eine natürliche Anschauung herrscht auch in der dumpfen Luft der Rechenklasse Büttners. Alle Zahlen von 1 bis 40 sollen seine Schüler zusammenzählen.

"Eine tüchtige Zeit werden sie dazu brauchen", denkt der Lehrer, "genug, um mich ein wenig von all der Mühsal ausruhen zu können." Kaum aber hat er sich mit seinem Prügelstock hingesetzt, als der kleine Gauß von seinem Platz aufspringt und nach vorn gelaufen kommt.

"Dar licht se", ruft er echt braunschweigisch dem verwunderten Lehrer glücklich zu und legt seine Tafel, wie es üblich ist, mit der Rechnung verkehrt auf das Pult.

"Nun", denkt Büttner, "eine schöne Dummheit wird das Burschchen in seinem Übermut begangen haben." Und spöttisch ruhen seine Blicke auf ihm, der siegesbewußt auf seinem Platze wartet.

Lange dauert es, bis die andern mühsam die 40 Zahlen zusammengebracht haben. Tafel nach Tafel wird auf die erste gelegt. Ironisch lächelnd dreht Büttner endlich wieder eine nach der andern um. Wie erstaunt er aber, als er zur ersten kommt und alf ihr nichts weiter steht als die eine richtige Zahl, die Zahl 820!

Ja, hier hat es kein ägerliches Addieren wie auf den andern Tafeln gegeben. Mit dem Blick eines frühen Meisters hat hier einer blitzschnell Zusammenhänge gesehen. Wie in einem lustigen Tanz haben sich ihm die 40 Zahlen des Lehrers geordnet. Ganz anders als zu Beginn standen sie plötzlich vor ihm, immer eine vom Anfang einer vom Ende zugesellt: Zwanzig Gruppen von je zwei Zahlen waren es schließlich, und immer ergaben die beiden zusammen 41. Was blieb also noch zu tun, als diese neue Zahl zu verzwanzigfachen? [3]

Der Lehrer Büttner ist erschüttert. Das Bürschchen da vor ihm hat also aus eigener Kraft das Gesetz zur Bildung der Summe einer arithmetischen Reihe entdeckt, ohne je etwas von einer solchen Reihe gehört zu haben. [4] Er läßt aus Hamburg ein Rechenbuch kommen: Remers Arithmetica. Das gibt er dem Jungen, damit er seinen Wissendurst besser stillen kann, als es in seiner Rechenklasse möglich ist.


[Note 3, p. 211]: 1 + 2 + 3 + 4 + ........ + 37 + 38 + 39 +40 wandelte sich dem Knaben um in 1 + 40/ + 2 + 39/ + 3 + 38/ + 4 + 37/ + ..........

[Note 4, p. 211]: Eine arithmetische Reihe wird durch eine Aufeinanderfolge von Zahlen gebildet, bei denen der Unterschied zweirer benachbarter immer der gleiche ist. Die Summe einer solchen Reihe wird nach folgender Formel berechnet:
S = n/2 (a + z).
Dabei bedeutet n die Anzahl der Zahlen, a die erste, z die letzte Zahl. Im Falle der Büttnerschen Aufgabe ist also n = 40, a = 1 und z = 40, daher
S = 20 (1 + 40) = 820.

Wußing, Hans. 1989. Carl Friedrich Gauss. Leipzig: BSB B. G. Teubner Verlagsgesellschaft. (pp. 9–11)

Gauß pflegte gegen Ende seines Lebens oft und gern aus seiner Kindheit zu erzählen. Auf diese Quelle geht eine Reihe von Anekdoten zurück, von denen man annehmen darf, daß sie dem Kerne nach unbedingt richtig sind.

Nach der einen berühmtgewordenen Geschichte hat der dreijährige Gauß mitgehört, wie sein Vater den Lohn für seine Gehilfen in der Gärtnerei berechnete. Im Begriffe, eine Summe auszuhändigen, wurde er mit dem Zwischenruf unterbrochen: "Papa, Du hast einen Fehler gemacht." Zur allgemeinen Verblüffung bestätigte eine Nachprüfung den Einwurf des Jungen. An dieser Stelle seiner Erzählung pflegte Gauß lachend hinzuzufügen, daß er eher rechnen als sprechen gelernt habe.

In Jahre 1784, also siebenjährig, gaben die Eltern ihren Sohn in die Katharinen-Schule, die von dem Lehrer J. G. Büttner betrieben wurde. Der erste Biograph von Gauß, Wolfgang Sartorius Freiherr von Waltershausen (1809–1876), Professor der Mineralogi und Geologie in Göttingen, hat eine Schilderung des damaligen Schulbetriebes gegeben, die auch für sich genommen recht interessant ist. Hier, in der Elementarschule, spielt auch die zweite berühmtgewordene Anekdote um den jungen Gauß.

"Es war eine dumpfe, niedrige Schulstube mit einem unebenen ausgelaufenen Fussboden, von der man nach der einen Seite gegen die beiden schlanken gothischen Thürme der Catherinen-Kirche, nach der anderen gegen Ställe und armselige Hintergebäude hinaus blickte. Hier ging Büttner zwischen etwa hundert Schülern auf und ab, mit der Karwatsche in der Hand, welche damals als ultima ratio (letztes Mittel – H. W.) seiner Erziehungsmethode von Gross und Klein anerkannt wurde und der nach Laune und Bedürfniss einen schonungslosen Gebrauch zu machen er sich berechtigt fühlte. In dieser Schule, die noch sehr den Zuschnitt des Mittelalters gehabe zu haben scheint, blieb der junge Gauss zwei Jahre ohne durch etwas Außerordentliches aufzufallen. Erst nach jener Zeit brachte es der Gang des Unterrichtes mit sich, dass auch er in die Rechenklasse eintrat, in welcher die Meisten bis zu ihrer Confirmation, bis etwa zu 15ten Jahre Blieben.

"Es ereignete sich hier ein Umstand, den wir nicht ganz unbeachtet lassen dürfen, da er auf Gauss' späteres Leben von einigem Einfluß gewesen ist und den er uns in seinem hohen Alter mit grosser Freude und Lebhaftigkeit öfter erzählt hat. Das Herkommen brachte es nämlich mit sich, dass der Schüler, welcher zuerst sein Rechenexempel beendigt hatte, die Tafel in die Mitte eines grossen Tisches legte; über diese legte der zweite seine Tafel u.s.w. Der junge Gauss war kaum in die Rechenklasse eingetreten, als Büttner die Summation eine arithmetischen Reihe aufgab. Die Aufgabe war indess kaum ausgesprochen als Gauss die Tafel mit den Tisch wirft: "Ligget se" (Da liegt sie). Während die anderen Schüler emsig weiter rechnen, multiplizieren und addieren, geht Büttner sich sich seiner Würde bewusst auf und ab, indem er nur von Zeit zu Zeit einem Mitleidigen und sarcastischen Blick auf den Kleinsten der Schüler wirft, der längst seine Aufgabe beendigt harte. Dieser sass dagegen rubig, schon eben so sehr von dem festen unerschütterlichen Bewusstsein durchdrungen, welches ihn bis zum Ende seiner Tage bei jeder vollendeten Arbeit erfüllte, dass seine Aufgabe sichtig gelöst sei, und dass das Rusultat kein anderes sein könne. Am Ende der Stunde wurden darauf die Rechentafeln ungekehrt; die von Gauss mit einer einzigen Zahl lag oben und als Büttner das Exempel prüfte, wurde das seinige zum Staunen aller Anwesenden als richtig befunden, währent viele der übrigen falsch waren und alsbald mit der Karwatsche rectificirt (berichtigt – H.W.) wurden."

Hier trat eine ungewöhnliche Begabung zutage: Im allgemeinen erzaählt man sich, Büttner habe die Aufgabe gestellt, alle ganzen Zahlen von 1 bis 100 aufzusummieren (gelegentlich wird auch von der Summe aller Zahlen von 1 bis 60 berichtet). Statt nun, wie ganz selbstverständlich für jeden normalen Schüler dieser Altersklasse, der Reihe nach zu rechnen: 1+2 = 3, 3+3 = 6, 6+4 = 10, 10+5 = 15, usw., fiel dem jungen Gauß auf, daß in der Summation 1 + 2 + 3 + ... + 97 + 98 + 99 + 100 jeweils aus zwei Zahlen am Anfang und Ende der Reihe die Zahl 101 zu bilden ist: 1+100 = 101, 2+99 = 101 usw. Es gibt 50 solcher Paare. Es bleibt also nur eine einfache Multiplikation zu erledigen: 101 × 50 = 5050. Kein Wunder, daß Gauß nur eine einzige Zahl auf die Tafel zu schreiben brauchte und die Lösung im Handumdrehen finden konne!

Zimmer, Harro. 1988. "Portrait eines Genies" Carl Friedrich Gauß, Fürst der Mathematiker. Transcript of a radio presentation: RIAS Berlin; Kulturelles Wort Wissenschaftsredaktion, Sendung vom 10.09.1988 (08.45 Uhr - 09.00 Uhr; RIAS I). Transcript available online. Link to PDF file (Viewed 2006-02-02)

Wie wäre es, verehrte Hörer, mit einer kleinen Rechenaufgabe? Mit einem Problem, das Sie ohne elektronischen Taschenrechner, nur mit Papier und Bleistift bewaffnet, lösen sollten. Zählen Sie bitte rasch die Zahlen von 1 bis 100 zusammen. Sie sollten dazu nicht mehr Zeit als 2 Minuten benötigen. Unmöglich?

Im Jahre 1786 stellte der Braunschweiger Schulmeister Büttner seinen 9- und 10-jährigen Schülern diese Aufgabe, um einmal eine Stunde Ruhe vor der lärmenden Rasselbande zu haben. Noch war er mit den Erläuterungen beschäftigt, als ein neunjähriger blonder Junge aufstand: "Liggetse"!

Mit diesen plattdeutschen Worten legte er seine Schiefertafel mit der Rückseite nach oben auf das Pult und setzte sich wieder auf seinen Platz. Schulmeister Büttner, der die Karwatsche - die neunschwänzige Katze - genau so virtuos zu handhaben pflegte wie die Kreide, hatte den vorwitzigen Burschen schon für eine gehörige Tracht Prügel vorgemerkt. Er wartete jedoch bis zum Ende der Stunde. Dann drehte er den Berg der vollgekritzelten Schiefertafeln um, so daß die zuerst abgegebene Tafel nun oben lag. Statt der erwarteten Zahlkolonnen sah er nur eine einzige mit Schönschrift geschriebene Zahl: 5050!

Zunächst mußten jene Schüler nach vorne treten, die ein falsches Ergebnis produziert hatten, und das waren nicht wenige. Je nach Größe des Fehlers ließ der gestrenge Herr Büttner die Karwatsche auf das Hinterteil der armen Sünder niedersausen. Dann rief er: "Gauß! Nach vorne kommen - Heraus mit der Sprache! Wie ist er zu diesem Resultat gelangt?" "Im Kopf ausgerechnet, Herr Lehrer. Für eine so einfache Aufgabe brauche ich Tafel und Griffel nicht!" Da war der Braunschweiger Schulmeister sprachlos: "Das muß er mir wohl näher erklären!" Der junge Gauß: "Nun ich habe mir überlegt, daß die erste Zahl 1 und die letzte Zahl 100 zusammen 101 ergeben. Das Gleiche gilt für 2 plus 99, für 3 plus 98, und so weiter. Das sind also 50 Zahlenpaare. Fünfzig mal 101 ergibt 5050!" Stiller, aber aufmerksamer Zeuge dieses Dialogs war ein junger Mann, Martin Bartels, der Gehilfe des Schulmeisters. Ihn interessierte weit mehr die hohe Wissenschaft als das simple ABC, und es dauerte nicht lange, bis das ungleiche Gespann - der neunjährige Schüler und der Hilfslehrer - in schöner Eintracht die komplizierten Werke der höheren Mathematik studierten.