My method for finding the initial guess is simply to find the closest perfect square and then estimating how far your number is from that perfect square. I mean, even from 1 to 10 this covers alot..... 1 4, 9, 16, 25, 36, 49, 64, 81, 100,

It's kind of easy for 4 digits because you just count by 10s, so 100s when squared 100, 400, 900 1600, etc

If you wanted to do a number like 550 or something then you are going somewhere between 400 and 900. (answer between 20 and 30 but closer to 20)

I mean I'm really overexplaining here because with your number , 3249, it doesn't take a lot of effort to see that 3600 is the closest perfect square to that number.

Since 3249 is smaller than 3600, you know your square root is going be smaller than 60. You can also somewhat tell that it is not a whole smaller than 60, and in fact if go down to the previous, of 2500, and try to guage where 3249 sits between 2500 and 3600, you would just apply a similar ratio between 50 and 60.

Half way between 2500 and 3600 would be about 3050 so you know 3249 is just over half way between 50 and 60.

I mean you can probably just look at 3249 and not think too hard to see that 56 is probably a fairly reasonable initial guess, but go ahead and try it and tell me how many iterations you need.