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

Question

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?

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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:

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

Answer 1

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.

Answer 2

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.)

Answer 3

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