What is the greatest trick in mathematics?

In elementary school in the late 1700’s, a class was asked to find the sum of the numbers from 1 to 100.  The question was assigned as “busy work” by the teacher, but one of them found the answer rather quickly by discovering a pattern.  His observation was as follows:

1 + 2 + 3 + 4 + … + 98 + 99 + 100

He noticed that if he was to split the numbers into two groups (1 to 50 and 51 to 100), he could add them together vertically to get a sum of 101.

1     + 2   + 3   + 4   + 5   + … + 48 + 49 + 50
100 + 99 + 98 + 97 + 96 + … + 53 + 52 + 51

1 + 100 = 101
2 + 99 = 101
3 + 98 = 101
.
.
.
48 + 53 = 101
49 + 52 = 101
50 + 51 = 101

He realized then that his final total would be 50(101) = 5050.

He was none other than "Carl Friedrich Gauss"
Thanks to him we now have a formula S=n(n+1)/2