Markland from The John Roan School, Gareth; Euen and Alex from Madras College, Scotland; and Chin Siang from Tao Nan School, Singapore; all sent in good solutions .
Jack from The Ridings High School described the pattern:
I noticed that the number of 4s and 8s each increased by 1 for each extra 6 and that the last digit was always a 9. I then predicted that 666667 ² would equal 444444888889 and I was correct. Therefore according to this pattern: (1 million 6s followed by a 7) ² would be written 1000001 4s followed by 1000000 8s followed by a nine.
67 ^{2}  =  4489 
667 ^{2}  =  444889 
6667 ^{2}  =  44448889 
66667 ^{2}  =  4444488889 
Doing these four calculations by long multiplication shows how this pattern works. If $m$ is the number of sixes in the number that is squared, the pattern is:
( $m$ sixes followed by $7$)$^2 =$ ($(m+1)$ $4$'s followed by $m$ $8$'s followed by a $9$).So
(one million sixes followed by a $7$)$^2 =$ (one million and one $4$'s followed by a million $8$'s followed by a $9$).
666  ...  666667  
666  ...  666667  


4666  ...  666669  (x7)  
40000  ...  000020  (x60)  
400000  ...  000200  (x600)  
.  
.  
.  
40  ...  000020  ...  000000  (x 6 x 10 ^{m 1} ) 
400  ...  000200  ...  000000  (x 6 x 10 ^{m} ) 


444  ...  444888  ...  888889  Total 