1. Choose a number with repeating 3's and a final 7.
2. The square is made up of:
   • the same number of 1's as there are repeating 3's in the number;
   • one 3
   • one fewer 5 than there are repeating 3's;
   • a final 69.

   Example:

1. If the number to be squared is 3337:
2. The square has:

   three 1's (same as
   repeating 3's)     1 1 1
one 3                       3
two 5's (one fewer than
   repeating 3's)             5 5
a final 69                        6 9

3. So the square of 3337 is 11,135,569.

   See the pattern?

1. If the number to be squared is 333337:
2. The square has:

   five 1's (same as
   repeating 3's)   1 1 1 1 1
one 2                         2
four 5's (one fewer than
   repeating 3's)               5 5 5 5
a final 69                              6 9

3. So 333337 × 3333337 = 111,113,555,569.