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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

Summing Up

Young Gauss’s “trick”...Click to Enlarge Image

As with the identity of the series, the details of how Gauss solved the problem remain a matter of conjecture. The algorithm that I suggested—folding the sequence in half, then adding the first and last elements, the second and next-to-last, etc.—is not the only possibility. A related but subtly different algorithm is mentioned by many authors. The idea is to write down the series twice, once forward and once backward, and then add corresponding elements. For the familiar series 1-100 this procedure yields 100 pairs of 101, for a total of 10,100; then, since the original series was duplicated, we need to divide by 2, arriving at the correct answer 5,050. The advantage of this scheme is that it works the same whether the length of the sequence is odd or even, whereas the folding algorithm requires some fussy adjustments to deal with an odd-length series.

A third approach to the summation problem strikes me as better still. The root idea is that for any finite set of numbers, whether or not the numbers form an arithmetic progression, the sum is equal to the average of all the elements multiplied by the number of elements. Thus if you know the average, you can easily find the sum. For most sets of numbers, this fact is not very useful, because the only way to calculate the average is first to calculate the sum and then divide by the number of elements. For an arithmetic progression, however, there is a shortcut: The average over the entire series is equal to the average of the first and last elements (or the average of any other elements symmetrically arrayed around the midpoint). If this was Gauss's secret weapon, then his mental multiplication was not 50 x 101 but 100 x 50½.

All three of these ideas—and a few more besides—have been presented by one author or another as the method that Gauss discovered during his first arithmetic lesson. Expressed as formulas for summing consecutive integers from 1 through n, the three rules (folding, double rows, average) look like this:

Three rulesClick to Enlarge Image

Mathematically, it's obvious they are equivalent: For the same value of n, they produce the same answer. But the computational details are different and, more important, so are the reasoning processes that lead to these formulas.

There is yet another way of thinking about the summation process: n(n + 1)/2 has been known since antiquity as the formula for the triangular numbers, those in the sequence 1, 3, 6, 10, 15, 21.... Thus some authors suggest that Gauss was thinking geometrically, forming an n-by-n + 1 rectangle and cutting it along the diagonal.

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