COMPUTING SCIENCE
Gauss's Day of Reckoning
A famous story about the boy wonder of mathematics has taken on a life of its own
Brian Hayes
Let me tell you a story, although it's such a well-worn nugget of
mathematical lore that you've probably heard it already:
In the 1780s a provincial German schoolmaster gave his
class the tedious assignment of summing the first 100 integers. The
teacher's aim was to keep the kids quiet for half an hour, but one
young pupil almost immediately produced an answer: 1 + 2 + 3 + ... +
98 + 99 + 100 = 5,050. The smart aleck was Carl Friedrich Gauss, who
would go on to join the short list of candidates for greatest
mathematician ever. Gauss was not a calculating prodigy who added up
all those numbers in his head. He had a deeper insight: If you
"fold" the series of numbers in the middle and add them in
pairs—1 + 100, 2 + 99, 3 + 98, and so on—all the pairs
sum to 101. There are 50 such pairs, and so the grand total is
simply 50×101. The more general formula, for a list of
consecutive numbers from 1 through n, is
n(n + 1)/2.
The paragraph above is my own rendition of this anecdote, written a
few months ago for another project. I say it's my own, and yet I
make no claim of originality. The same tale has been told in much
the same way by hundreds of others before me. I've been hearing
about Gauss's schoolboy triumph since I was a schoolboy myself.
The story was familiar, but until I wrote it out in my own words, I
had never thought carefully about the events in that long-ago
classroom. Now doubts and questions began to nag at me. For example:
How did the teacher verify that Gauss's answer was correct? If the
schoolmaster already knew the formula for summing an arithmetic
series, that would somewhat diminish the drama of the moment. If the
teacher didn't know, wouldn't he be spending his interlude
of peace and quiet doing the same mindless exercise as his pupils?
There are other ways to answer this question, but there are other
questions too, and soon I was wondering about the provenance and
authenticity of the whole story. Where did it come from, and how was
it handed down to us? Do scholars take this anecdote seriously as an
event in the life of the mathematician? Or does it belong to the
same genre as those stories about Newton and the apple or Archimedes
in the bathtub, where literal truth is not the main issue? If we
treat the episode as a myth or fable, then what is the moral of the story?
To satisfy my curiosity I began searching libraries and online
resources for versions of the Gauss anecdote. By now I have over a
hundred exemplars, in eight languages. (The collection of versions
is available here.) The sources range from
scholarly histories and biographies to textbooks and encyclopedias,
and on through children's literature, Web sites, lesson plans,
student papers, Usenet newsgroup postings and even a novel. All of
the retellings describe what is recognizably the same
incident—indeed, I believe they all derive ultimately from a
single source—and yet they also exhibit marvelous diversity
and creativity, as authors have struggled to fill in gaps, explain
motivations and construct a coherent narrative. (I soon realized
that I had done a bit of ad lib embroidery myself.)
After reading all those variations on the story, I still can't
answer the fundamental factual question, "Did it really happen
that way?" I have nothing new to add to our knowledge of Gauss.
But I think I have learned something about the evolution
and transmission of such stories, and about their place in the
culture of science and mathematics. Finally, I also have some
thoughts about how the rest of the kids in the class might have
approached their task. This is a subject that's not much discussed
in the literature, but for those of us whose talents fall short of
Gaussian genius, it may be the most pertinent issue.
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