# Did Gauss find the formula for $1+2+3+\ldots+(n-2)+(n-1)+n$ in elementary school?

I heard Gauss's primary school teacher gave some busy-work to his class: to add all the numbers between 1 and 100 up. Gauss immediately wrote 5050. His teacher was shocked, so she told him to add up all the numbers to 1000. And just as quickly he wrote 500500.

Did Gauss derive the $1+2+3+\ldots+(n-2)+(n-1)+n=\frac{n(n+1)}{2}$ formula in elementary school? If so, what biography of Gauss discusses it?

His proof:

$1+2+3+\ldots+(n-2)+(n-1)+n$

$+$

$n+(n-1)+(n-2)+\ldots+3+2+1$

$=$

$\underbrace{(n+1)+(n+1)+(n+1)+\ldots+(n+1)+(n+1)+(n+1)}_{n}=n(n+1)$

And then he divided by $2$ because of double-counting to get:

$\frac{n(n+1)}{2}$

• It's worth mentioning that there are at least 135 different versions of this story: bit-player.org/wp-content/extras/gaussfiles/gauss-snippets.html. It's quite possibly true but a good answer should say which version is correct and give credible sources, which may be quite difficult. Commented Nov 11, 2014 at 23:05
• This question was originally asked on MSE and was moved on my recommendation. Commented Nov 11, 2014 at 23:46
• This widely known story sounds very plausible. But how can you prove or disprove this story? You don't have to be a Gauss to do this. Commented Nov 12, 2014 at 0:19
• @HDE226868 I think that's a bad idea, cf. the discussion on tags we had on meta. No name-tags!
– Danu
Commented Nov 12, 2014 at 0:20
• See here for a discussion of the source of the anecdote, and see here, page 4, for the source (1856). Commented Nov 12, 2014 at 12:37

According to an American Scientist article (Gauss' day of reckoning by Brian Hayes, Volume 94 p. 200) mentioned in the comments, the original source for this story, or at least a story very similar to it, was Gauss zum Gedächtnis, a memorial written very soon after Gauss' death by Wolfgang Sartorius, a colleague of his at Göttingen (however, I am not sure if Gauss and Sartorius personally knew each other). Sartorius claims that Gauss was fond of recounting the tale himself in his old age, and indeed specifically mentions that the problem was "the summing of an arithmetic series", without mentioning which arithmetic series.

According to the American Scientist article, the first appearance of the 1-100 summation in this anecdote is in a biography of Gauss by Ludwig Bieberbach in 1938, which simultaneously introduces the story about Gauss' rearrangement method for summing the numbers.

Now, it could be that Bieberbach, the person who Bieberbach learned the story from, or the person they learned the story from, or somebody else in the 80-year long chain between Sartorius and Bieberbach made up the story about the 1-100 sum, or somehow mistakenly inserted it into the anecdote. However, it could also be the case, if Gauss was indeed as fond of telling the story as Sartorius claims, that Sartorius simply wasn't aware of the 1-100 detail, or felt it wasn't important enough to include in his biography, but that the fact survived in oral tradition among other German mathematicians until Bieberbach wrote it down in 1938. It's impossible to know.

However, it would seem very odd for a primary school teacher to set a problem involving the summing of an arithmetic series that was much stranger than "the numbers from one to a hundred". The purpose of the exercise was clearly to test the students' ability at large-scale, repetitive computation. It seems like summing the numbers from 1-100 would be sufficient, and something more complicated like "sum every 11th numbers from 344 to 700" or whatever would seem unnecessary.

• This site is interesting artofproblemsolving.com/Resources/articles.php?page=gallery Commented Jan 3, 2015 at 9:47
• Some additional information: Brian Hayes continues to update his list of tellings of the story. (This is the same as the link in Logan M's comment.) The most important addition to the list: Hayes has discovered a version by Franz Mathé that predates Bieberbach's by 30 years and contains both the 1-100 detail and the pairing method. Commented Jan 25, 2015 at 3:28
• Another piece of information: it appears that Sartorius and Gauss were very close friends. Helen Worthington Gauss in the preface to her 1966 translation of Gauss zum Gedächtnis writes that Sartorius "wrote over 100 years ago and immediately after the death of his long-cherished friend and colleague. This to some extent explains the extreme feeling and language of the Memorial." This is borne out in the text. ... Commented Jan 25, 2015 at 4:21
• ... We find, for example, this passage: "One day in the winter of 1832 I happened in at the Observatory. Always ready to teach and share his thoughts, Gauss picked up a small box-compass and showed me how the iron rods which closed the window were themselves turned into magnets through the influence of the earth's magnetism." Sartorius, a geologist who would have been only about 22 years old at the time, carried out research on geomagnetism a few years later. ... Commented Jan 25, 2015 at 4:22
• ... Toward the end of Gauss's life, it appears they were intimate. Sartorius writes movingly of Gauss's final days. For example, "With the ups and downs of his illness I did not see Gauss again until January 14th. ... I found Gauss weaker but cheerful. He related some incident out of his earlier life; his blue eyes sparkled, the last time I saw them so." The writing continues in the same vein for several pages. Commented Jan 25, 2015 at 4:22

Although it's an old question, but perhaps this insight would be beneficial to those stumbling on this question.

Gauss wasn't the first to derive this formula per se.

Algebraically, you'd see "how" to derive the formula by working through Gauss' derivation. Purely by observation you'd see that you can pair up the extreme values to get the same sum and just count/multiply them out and divide the result by two.

However, this formula was well known in ancient Greece too. Pythagoreans had a habit of "visualizing" numbers by arranging discs/stones in particular patterns.

A triangular number was that which could be arranged in a triangle:

*   *      *
* *   *   *
*  *  *
1   3      6


A square number was that which could be arranged in a square (hence our usage of $x$ squared):

 *  * *   * * *
* *   * * *
* * *
1   4      9


Now, if you see carefully, $6 = 1 + 2 + 3$ . By extension $10 = 1 + 2 + 3 + 4$ and so on.

However, you can easily see the relation between a triangular number and a corresponding square number, by doubling the triangle

* + + + +
* * + + +  => 2 x T(10) = S(4) + 4 = 4 x 5
* * * + +
* * * * +


Doubling gives you a rectangle - you can get back the triangle simply by observing that

$2 \cdot T(n) = n(n+1) => T(n) = \frac{n(n+1)}{2}$

Triangular number just happen to arrange themselves in a simple pattern of an arithmetic progression of consecutive numbers.

What Gauss did was think of this algebraically. We credit Gauss with this story, but there isn't a reason to. I'd say we look at the story from the POV of a "smart kid" who may or may not have known about the Greek history of numbers but was able to "sit back and think" vs. attacking the problem in a brute force fashion.

(Side note: The formula for sum of squares took some work and isn't really immediately apparent. I believe Archimedes derived it successfully.)

• "However, this formula was well know in ancient Greece too. Pythagoreans had a habit of "visualizing" numbers by arranging discs/stones in particular patterns." Do you have a source for this? If not, then this post is almost entirely just an explanation of the sum in this anecdote, and not an answer to the original query. Commented Feb 13, 2018 at 22:58
• For what it's worth, in my own experience, back in the pathetic pre-internet days, when all we had to play with was sticks, mud, rocks, and math, at a very mathematically-naive point in my life, I did hit upon some of these formulas, indeed by manipulating geometric shapes, as I imagine many (interested/bored) people had done for thousands of years. Commented Feb 14, 2018 at 1:12
• @BrendanW.Sullivan - I'll do my best to dig out a reference. I learnt of this as a result of going through the history of math. I just don't remember where off the top of my head. Will add it once I find it. FWIW, even without a reference, this proof should've been easy for them to just "see". I believe Archimedes makes a reference to this when searching for the formula for the sum of squares. But either way I'll have to find the source.
– PhD
Commented Feb 14, 2018 at 19:26
• @BrendanW.Sullivan - On page 14 of Calculus Reordered, David Bressoud claims Archimedes used the sum to derive a formula for the sum of squares in On Spirals. The origin of the sum itself is "lost in the mists of time" (page 199). Commented Jan 19, 2021 at 19:27
• @DavidM - what a wonderful reference! I just ordered a copy - thank you :)
– PhD
Commented Jan 20, 2021 at 20:49

The potato in the ring and the n-ary number.

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