Solution:

Here's an elegant way:

1 + 36 = 37
2 + 35 = 37
3 + 34 = 37
.....................
.....................
18 + 19 = 37

Adding the above equations gives

1 + 2 + 3 + ....... + 36 = 18 × 37 = 666

Food for thought:

The general formula is:

1 + 2 + 3 + ....... + n = n (n + 1) / 2.

Is the formula valid when n is an odd integer?