<s>[INST] Calculation of a square root of a big number 
How can I calculate the following number: 
  $$ \sqrt{444 \cdots (2n \text{ digits}) + 111 \cdots (n+1 \text{ digits}) - 666 \cdots (n \text{ digits})}.$$ 

My trying : 
I have tried to calculate these data by observing the pattern of $ 7^2 , 67 ^2 ,667^2 , 6667^2 , \dots$. But there is no promising news to solve it by this process. 
 [/INST] Hint:
Use the fact that $111\dots 111 = \frac{10^n-1}{9}$ (where there are $n$ ones)
Then your number is 
$$4\frac{10^{2n}-1}{9} + \frac{10^{n+1}-1}{9} - 6\frac{10^n-1}{9} = \frac{4\times 10^{2n}-4 +10^{n+1}-1-6\times 10^n+6}{9}$$
$$=\frac{4\times 10^{2n}+4\times 10^n+1}{9}$$
$$= \frac{(2\times 10^n+1)^2}{9}$$
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