1. Choose a number with a 1 and repeating 9's.
2. The square is made up of:
   • first digits: 3 & one fewer 9 than repeating 9's
   • next digits: 6 & one fewer 0 than repeating 9's
   • last digit: 1

   Example:

1. If the number to be squared is 1999:
2. The square has:

   first digits: 3 and one fewer
    9 than 9's                      3 9 9
next digits: 6 and one fewer
    0 than 9's                            6 0 0
last digit: 1                                   1

3. So 1999 × 1999 = 3996001.

   See the pattern?

1. If the number to be squared is 199999:
2. The square has:

   first digits: 3 and one fewer
   9 than 9's                 3 9 9 9 9
next digits: 6 and one fewer
    0 than 9's      &                   6 0 0 0 0
last digit: 1                                     1

3. So 199999 × 199999 = 39999600001.