For even number of digits of the number to find the square root of:

3672

average the number you get when you slide the decimal point halfway across and the power of 10 you moved the decimal point by:

(36.72 + 100)/2 = 68.36, this will always be a bit larger than the actual square root, in this case 60.59

For odd number of digits first multiply by 10...

36725 x 10 = 367250

Then proceed as before

(367.250 + 1000)/2 = 683.625 and then divide by sqr(10) = 216, the actual is 191

It works even better in binary...

36725=

1000111101110101

10001111

2^7= 1000000

(128 + 241)/2 = 181 when the actual is 191...

So this is a good way to find the initial seed value to continue with a precise method that needs a good guess like the Babylonian method to work it's best...

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