Gauss Again

Remember that story about Gauss, the brilliant 18th-century mathematician? As a schoolboy, he showed up his teacher when he tried to give the class a make-work problem involving adding up all the numbers from 1 to 100, and Gauss solved it in just a few moments? I do. I first heard it somewhere in elementary school, and it made me feel incredibly inadequate. (I knew I'd never make it to junior high school.)

Turns out, however,that the story's a very old urban legend. Brian Hayes, writing in American Scientist Online, has collected over 100 examples of the story in eight languages. According to Hayes, the earliest account of the story, though able to claim Gauss himself as its source, was written long after the fact and does not specify what arithmetic problem was assigned to the class. In other words, the anecdote simply says Gauss solved an unnamed problem very quickly. It was later authors who inserted the detail about the problem being to add up a series of numbers from 1 - 100. Hayes' account of what he's learned about the history of the story and its evolution is worth reading; more interesting still is his point that the other students -- the putative dunces who couldn't see a shortcut to adding up a series of integers -- would have found lesser shortcuts of their own if they actually tried adding up all those numbers, because they would've seen obvious patterns. As he writes,

Let me invite you to take a sheet of paper and actually try adding the numbers from 1 to 100.

Finished? Already?

All right, all right, no need to show it to me. So you've guessed the next part of what Hayes says:

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?

Which brings us to Hayes' real point:

On first hearing this fable, most students surely want to imagine themselves in the role of Gauss. Sooner or later, however, most of us discover we are one of the less-distinguished classmates; if we eventually get the right answer, it's by hard work rather than native genius. I would hope that the story could be told in a way that encourages those students to keep going. And perhaps it can be balanced by other stories showing there's a place in mathematics for more than one kind of mind.

... and in many other fields as well.