1. Choose a a number made up of nines (up to nine digits).
2. The answer will have one less 9 than the number, one 8,
   the same number of zeros as 9's, and a final 1

   Example:

1. If the number to be squared is 9999
2. The square of the number has:

   one less nine than the number  9 9 9
one 8                                8
the same number of zeros as 9's        0 0 0
a final 1                                    1

3. So 9999 × 9999 = 99980001.

   See the pattern?

1. If the number to be squared is 999999
2. The square of the number has:

   one less nine than the number  9 9 9 9 9
one 8                                   8
the same number of zeros as 9's           0 0 0 0 0
a final 1                                           1

3. So 999999 × 999999 = 999998000001.

This is not a very demanding mental math exercise, but it is an interesting pattern.