> anyone else whose jaw dropped

Wait, really? You never worked out in school (or on your own before it ever came up in school) that

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

and likewise that

  2 + 4 + 6 + ... + 2n = n * (n + 1)

and

  1 + 3 + 5 + ... + (2n - 1) = n * n

? (of course, it also comes up geometrically if you start looking at the number of little squares in a big square, and count the next row)

It always struck me as obvious; with a square of n*n, just add n + n (for the two sides) + 1 (for the corner) and you get the next square. 2n+1 progresses 1, 3, 5 etc., plain as day.

What got me thinking about it (at perhaps 11 or 13 or so) was trying to turn it into a way of calculating the square root.

The visualization makes it seem obvious. Thanks.