Here's my working for that problem.

Because it's a puzzle, it will be in the form sqrt(x^2), so the answer will be x.

The only perfect square I know for this level that could match is (a+b)^2 = a^2 + 2ab + b. With b=1, the first term needs to be in the form a^2 + 2a or a(a+2).

I had no intutition for what this could be, then noticed it needs to be close to a perfect square itself, and this closest to this would be the two inner factors times the two outer factors (for it to work out neatly). Writing as binomils for ease of calculation:

  ((1000+0)(1000+3))   ((1000+1)(1000+2))
  (1,000,000 + 3000)   (1,000,000 + 3000 + 2)

This is the form sought! (with a=1,000,000 + 3000). It works because 0+3=1+2. So we have

  sqrt( (1,000,000 + 3000) + 1)^2 ) = 1,000,000+3000+1 = 1,003,001