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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

Doing It the Hard Way

So much for how the prodigious Carl Friedrich Gauss solved the problem. What about the rest of the students in the class? Let me invite you to take a sheet of paper and actually try adding the numbers from 1 to 100.

Finished? Already?

What I discovered when I tried this experiment is that it's really hard to do it the hard way. You may set out to plod dutifully through all the addition operations, but shortcuts present themselves even when you're not looking for them. Suppose you adopt the standard primary-school algorithm, writing down all 100 numbers in a tall column and then starting work on the units digits. After the first 10 digits, the partial sum is 45; the next 10 digits add another increment of 45, bringing the partial sum to 90; then 45 more makes 135, and so on. How far would a student get in this process before recognizing a repetitive pattern? On turning to the tens digits, the pattern is even harder to miss: There are ten 1s followed by ten 2s, then ten 3s, etc. Surely any student who has the skills to complete this task at all would not add those repeated numbers one by one. A more likely strategy would be the one Sartorius implied when he wrote "count, multiply and add"—the phrase that Helen Worthington Gauss reduced to mere "adding."

On a small slate or a sheet of paper, it's difficult to write 100 numbers in a column, and so students would likely break the task down into subproblems. Suppose you start by adding the numbers from 1 to 10, for a sum of 55. Then the sum of 11 through 20 is 155, and 21 through 30 yields 255. Again, how far would you continue before spotting the trend?

Admittedly, these shortcuts can't match the elegance and ingenuity of Gauss's method. They are tied to the decimal representation of numbers, and they also don't generalize as well to arithmetic progressions other than lists of consecutive integers. But they do remind us that there's usually more than one good way to solve a problem.

I suspect that only one kind of student would ever be likely to add the numbers from 1 through 100 by performing 99 successive additions—namely, a student using a computer or a programmable calculator. And for that student, the simplest strategy might in fact be the best one.

We can hope that a modern Büttner—deprived of his whip, of course, and teaching in a classroom where computers have replaced slates—would not be drilling students on skills of such dubious utility as adding up a long series of numbers by hand. But the new Büttner just might ask his pupils to write a program to calculate the sum of any arithmetic progression. A new Gauss, with the same keen insight, could create a very efficient program based on the pairing idea—and that feat still deserves the highest admiration. But the modern Gauss might not be the first to fling his or her laptop on the table and cry "There it lies!" Writing that clever program—and testing and debugging it, and proving its correctness—would be no quicker than writing the straightforward step-by-step version. In this respect, technology may be something of an equalizer.

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A collection of versions of the Gauss anecdote

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