Segmented double torus

[mklotz]

Reminds me of one of my favorite mathematics anecdotes.

Karl Gauss was a mathematical prodigy, yet he was required to go to the German equivalent of grade school. One day the class was noisy and the teacher, as punishment, told them to get out their slates and add all the numbers from 1 to 100.

Being good little German kids conditioned to respond to authority, they all got to work. All, that is, except little Karl who scratched something on his slate, sat back and lounged comfortably in his seat.

The teacher confronted him and asked why he wasn't working. Little Karl said, "I'm done" and held up his slate with the correct answer, 5050, written on it.

How could you have done it so quickly, asked the teacher.

Well,

1 + 100 = 101
2 + 99 = 101
...
50 + 51 = 101

so the answer must be 50 * 101 and I did that in my head.

Unwittingly, little Karl had devised the formula for the sum of the integers.

History does not record what the teacher said at this point. Being of German descent, I can guess but we'll leave that for another time.