1. Choose a number with repeating 6's and a final 9.
2. The square is made up of:
   • the same number of 4's as there are repeating 6's;
   • a 7
   • one fewer 5 than there are repeating 6's;
   • A final 61.

   Example:

1. If the number to be squared is 6669:
2. The square has:
   

                           same number of 4's as repeating 6's: 4 4 4
                           a 7                                        7
                           one fewer 5 than repeating 6's               5 5
                           a final 61                                      6 1
                           

3. So the square of 6669 is 44,475,561.

   See the pattern?

1. If the number to be squared is 666669:
2. The square has:
   

                           same number of 4's as 6's: 4 4 4 4 4
                           a 7                                  7 
                           one fewer 5 than repeating 6's         5 5 5 5
                           a final 61                                     6 1
                           

3. So 666,669 × 666,669 = 44,447,555,561.