All right, here’s a fun one. It usually comes as part of a story. The story as told is mostly true, though a few details have been a little fudged by the winds of history. It goes like this: when the young Carl Gauss was a small child in school, his teacher wished to kill some time by making the students practice their addition by adding the integers from 1 to 100. Gauss worked for about 10 seconds and turned in the correct answer. His method was exceptionally clever. By doing the problem twice he made the calculation much easier.
First he wrote down the problem, calling the unknown solution N:
Then he did it again, backwards:
It’s easy to add the two together. Just do it term by term: 1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101, etc. The two equations are just a string of that term repeated 100 times. As such, we have this:
100 times 101 is 10100. Divide that by 2, and the answer is 5050. Game, set, and match Carl Gauss. The same procedure works just fine for adding to even higher numbers. If you’re adding the integers from 1 to n, the same procedure gives you a Sunday Function that will do it in a heartbeat:
If you want to add the integers from one to a million, this will give you the answer a heck of a lot faster than actually sitting down and punching a million numbers into your calculator.
In the grand scheme of things this little trick is the very least of what we owe Gauss. But it’s no less a useful thing to know, and I’ve seen it more than once in useful mathematical physics.