Gauss's Schoolboy Trick

A famous story tells how a young Carl Friedrich Gauss, told to add every number from 1 to 100, answered in seconds: 5,050. The classroom details are likely embellished, but the method is real. Pair the first and last numbers (1+100=101), the second and second-to-last (2+99=101), and so on. There are 50 such pairs, so 50 x 101 = 5,050.

This is the arithmetic series formula: n(n+1)/2. It works for any sequence with a constant step. Want the sum of even numbers from 2 to 200? That is 100 terms of step 2: 100 x 202 / 2 = 10,100. One insight — pairing the extremes — replaces adding long columns.