Jumping Jack Flash (Arithmetic)

It's a Gauss, Gauss, Gauss.

Everybody learns the story of Gauss in their high school pre-calc course. Some schoolteacher hundreds of years ago decided he or she wanted an afternoon of quiet in the classroom — maybe to grade some papers, maybe to just steal a short break from having to interact with the little hellions. Remember, this was long before in-service days were invented.

So the teacher orders the class to add up all the numbers from 1 to 100, and the students begin furiously inking out their calculations: 1 + 2 = 3, 3 + 3 = 6, 6 + 4 = 10, and so on.

Except for one student. Let's call him Karl Gauss, for that was his name. Karl was a very bright child, and at the tender age of three had already begun finding and correcting payroll errors in his father's ledger. Karl was struck by a curious fact about the nature of the problem of adding a consecutive series of numbers. Namely, he spotted a pattern.

Karl Gauss noticed the opposite ends of the sequence always equaled the same value. 1 + 100 = 101. 2 + 99 = 101. 3 + 98 = 101. Karl realized that he could, theoretically, write the numbers 1 through 100 on a big long line, and then write a line of the numbers 100 through 1 directly beneath the first. That would give him 100 additions, each the sum of 101. So 100 times 101 is the result of adding the numbers 1 to 100 twice. Karl only had to divide (100 x 101) by 2 in order to find the answer the teacher wanted.

So Karl, brilliant as he was, figured all of this in his head. He wrote "5050" down on his paper and turned it in immediately. It was very likely the kind of spectacle we now refer to as "shock and awe".

I bet the teacher let him go home early after that little stunt.

But that's not my point. The point is that we still attribute this rapid addition technique to Gauss today, even though he first used it for that hallmark "5050" application some time in the 1780s. Hundreds of years later, students learn Gaussian summation and they hear the story of how Karl Gauss was so much smarter than you or me.

It is a wonder that Pythagoras, centuries upon centuries before Gauss, came so close to developing the same formula and, apparently, didn't. Pythagoras was obsessed with triangles, and so he was intimately versed in the idea of triangular ordering: one barrel sits alone, one barrel sits atop of two, one barrel sits atop two and those two sit atop three, and so on. Pythagoras knew this progression, and he knew the series that resulted from it: 1, 3, 6, 10, 15, and so forth. This is the same series Gauss solved in 60 seconds. Now, I'm not saying Pythagoras was a dimwit for missing this, but the 100th number in the series called the "triangular numbers", is 5050. I hesitate to think that Pythagoras wouldn't have put together a quick way to calculate the number of barrels stacked into a triangle x-many rows high.

I mean c'mon. It's Pythagoras. He had to have figured this out, right? Mathematics has some infamously weird syntax and methods of giving credit. Fermat developed Fermat's Little Theorem, but good luck deducing who first came up with the Chinese Remainder Theorem. I'd like to know if Gaussian summation was something that Pythagoras knew, and taught his followers, and which died with them when his math cult was disbanded.

Imagine it: Pythagoreas knowing the secrets of rapid addition in 500 BC. That is as close as you can get to, to quote one of my favorite movies, "finding a 747 made a thousand years before the Wright brothers flew". Only in this case, it's finding a rapid way to add integer series two thousand years before Karl Gauss got out of school early.

No wonder Pythagoras was so secretive. That kind of thing back then was a full-on "we hunt you down and kill you for the knowledge in your brain" sort of situation. Come to think of it, I'm pretty sure that's how Pythagoras died.

He knew Gaussian sums. Woo! Look at me, ma! I'm like a math historian over here!