I’m not sure how old I was when I heard the anecdote about Carl Friedrich Gauss, but he was in primary school when the event occurred, and I recall admiring his cleverness and feeling a little humbled, so I must have been older when I heard about him than he had been when he solved.

But that’s the difference between a math prodigy and the kind of bright child I was. I got him, but I wasn’t him.

It was probably around 1787. The story was that he misbehaved and his teacher tried to put him in his place by having him add all the integers from 1 to 100. Gauss replied with “5050” in a matter of seconds. The way I understood it, he saw the problem like a number line in the air. One was at the left end and 100 at the right end, and 50 was right smack in the middle. He immediately envisioned connecting the 1 to the 99 and the 2 to the 98 and the 3 to the 97, and so on. It was obvious that he would acquire 49 hundreds that way, and that the line would still have, unused, that 100 at the end and that 50 in the middle. Thus: 5050.

I love this story because it so sweetly demonstrates how the way you view a problem can lead to an elegant or cumbersome solution. So I looked it up before writing it up.

The first link was to wikipedia. I went there. And not only found the anecdote but even the teacher’s name (J.G. Büttner). I was surprised, however, to read this: “Gauss’s presumed method was to realize that pairwise addition of terms from opposite ends of the list yielded identical intermediate sums: 1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101, and so on, for a total sum of 50 x 101 = 5050.”

Now the reference goes on to say details of the story are not certain, and clearly we’re not seeing clearly into Gauss’s brain, and multiplying by 101 does lead more easily to the general formula for the sum of an arithmetic sequence, but what do you think?

Do you imagine young Carl Friedrich Gauss added hundreds and then adjusted, or collected 101’s and then multiplied?