1. Choose a number with repeating 6's and a final 3.
2. The square is made up of:
   • one fewer 4 than there are repeating 6's
   • 39
   • same number of 5's as 4's in the square
   • a final 69

   Example:

1. If the number to be squared is 6663:
2. The square has:

   two 4's (one fewer than
      repeating 6's)           4 4
Next digits: 39                    3 9
two 5's (same number as repeating 6's) 5 5
A final 69                                 6 9

3. So the square of 6663 is 44,395,569.

   See the pattern?

1. If the number to be squared is 666663:
2. The square has:

   four 4's (one fewer than
      repeating 6's)   4 4 4 4
Next digits: 39                3 9
four 5's (same as repeating 6's)   5 5 5 5
A final 69                                 6 9

3. So the square of 666663 is 444,439,555,569.