The Square Root of 4444222225
=============================
Let us find a positive x, such that:

  x² = 4444222225

without a calculator or the pen-and-paper algorithm.
The first step will be to look at its digits:

Four fours, then four twos and then 25.
Or
  4444222225 = 4444000000 + 222200 + 25 =
             = 4444⋅10⁶ + 2222⋅100 + 25 =
	     = 4⋅1111⋅10⁶+ 2·1111·100 + 25 =

               4⋅9999⋅10⁶+ 2·9999·100 + 225
             = ---------------------------- =
	                   9

               400(10⁴-1)10⁴ + 200(10⁴-1) + 15²
	     = -------------------------------- 
	                     9

Let a=10⁴-, then

             400a(a+1) + 200a + 15²
4444222225 = ---------------------- =
                       9

             400a² + 600a + 15²
           = ------------------ =
	             9

             (20a)² + 2·15(20a) + 15²
	   = ------------------------ =
	               9

             (20a + 15)²   (20·9999 + 15)²   
	   = ----------- = --------------- = (20·3333 + 5)² =
	          9              3²

           = 66665²
	   
Thus,

  +-----------+
  | x = 66665 |
  +-----------+