You know the story about Gauss? If not, he was about 11 or 12, and the teacher asked the boys to evaluate the sum 1+2+3+...100.

3) using simple but mind-numbing algebra.

The boys all did it the long way and came to various different answers. Gauss said 5050, which was right. The teacher asked how he did it so quickly. Gauss replied that you could split the sequence into pairs: (1+100) + (2+99) + ... (50+51). 50 such pairs, each the same value 101, 50 x 101 = 5050.

The second method is an example of mathematical insight, but hard to encode 'insight'.