When the great mathematician Carl Friedrich Gauss was a young lad in elementary school, the teacher posed a problem for the class: find the sum of the first 100 numbers. Gauss almost immediately came up with the answer: 5,050. He was able to do this not by rapid addition of the numbers 1-100, but by means of a sudden insight. Visualize if you will, along with the young Gauss, the numbers arrayed in the following manner:
1 + 2 + 3 . . . 50 + 51 . . . 98 + 99 + 100
Notice that the first and last numbers, 1 and 100, add up to 101.
The second and next-to-last numbers, 2 and 99, also add up to 101.
The same holds for all the successive pairs of numbers until you reach 50 and 51 whose sum is 101.
Therefore, as Gauss saw in a twinkling of the eye, 50 pairs of numbers, each pair adding up to 101, is 50 x 101 — or 5,050. How clever is that?
We may surmise that, upon revealing his method, both the teacher and the young Gauss enjoyed a meeting of the eyes — and an intellectual smile.
HyC