A teacher once asked her class to sum the integers from one to 100. The young Karl Freiderich Gauss--one of mankind's truly great mathematicians--promptly strode to her desk and slapped his small chalkboard on the desk. "There it lies!" He exclaimed.
The correct answer was indeed there. 5,050.
How did he do it?
He realized that you can pair up numbers like so:
0 1 2 3 4 5 6 7 8 9.......99 100
100 99 98 97 96 95 94 93 92 91.......01 0
The total of all the numbers in the two rows is 10,100 (100 times 100 plus 50 plus 50). But the second half of the sums is redundant (those greater than 50), so the correct answer is one-half of 10,100 or 5,050.
--Boris