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Gauss's Day of Reckoning

A famous story about the boy wonder of mathematics has taken on a life of its own

Brian Hayes

Making History

If Sartorius did not specify a series running from 1 to 100, where did those numbers come from? Could there be some other document from Gauss's era that supplies the missing details? Perhaps someone to whom Gauss told the story "with amusement and relish" left a record of the occasion. The existence of such a corroborating document cannot be ruled out, but at present there is no evidence for it. None of the works I have seen makes any allusion to another early source. If an account from Gauss's lifetime exists, it remains so obscure that it can't have had much influence on other tellers of the tale.

In the literature I have surveyed, the 1-100 series makes its first appearance in 1938, some 80 years after Sartorius wrote his memoir. The 1-100 example is introduced in a biography of Gauss by Ludwig Bieberbach (a mathematician notorious as the principal instrument of Nazi anti-Semitism in the German mathematical community). Bieberbach's telling of the story is also the earliest I have seen to specify Gauss's strategy for calculating the sum—the method of forming pairs that add to 101. Should Bieberbach therefore be regarded as the source from whom scores of later authors have borrowed these "facts"? Or is this a case of multiple independent invention?

If you think it utterly implausible that two or more authors would come up with the same example and the same method, then Bieberbach himself is disqualified as the source. A full millennium before Gauss and Büttner had their classroom confrontation, essentially the same problem and solution appeared in an eighth-century manuscript attributed to Alcuin of York.

A catalog of stories...Click to Enlarge Image

Furthermore, in the years since Bieberbach wrote, there is unmistakable evidence of independent invention. Not all versions agree that the sequence of numbers was the set of consecutive integers from 1 through 100. Although that series is the overwhelming favorite, many others have been proposed. Some are slight variations: 0-100 or 1-99. Several authors seem to feel that adding up 100 numbers is too big a job for primary-school students, and so they trim the scope of the assignment, suggesting 1-80, or 1-50, or 1-40, or 1-20, or 1-10. A few others apparently think that 1-100 is too easy, and so they give 1-1,000 or else a series in which the difference between successive terms is a constant other than 1, such as the sequence 3, 7, 11, 15, 19, 23, 27. (The example series chosen by various authors and other features of the versions are tabulated in the table at right.)

Perhaps the most influential version of the story after that of Sartorius is the one told by Eric Temple Bell in Men of Mathematics, first published in 1937. Bell has a reputation as a highly inventive writer (a trait not always considered a virtue in a biographer or historian). He turns the Braunschweig schoolhouse into a scene of gothic horror: "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." Very cinematic! When it comes to the arithmetic, however, Bell is one of the few writers who scruple to distinguish between fact and conjecture. He doesn't claim to know the actual numerical series, but writes: "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." (Personally, I'd have a hard time even writing that problem on a small slate, much less solving it.)

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Related Internet Resources

A collection of versions of the Gauss anecdote

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