1. Choose a number with a 4 and repeating 9's.
2. The square is made up of:
   • first digits: 24 & one fewer 9 than repeating 9's in the number
   • next digits: same number of 0's as repeating 9's in the number
   • last digit: 1

   Example:

1. If the number to be squared is 4999:
2. The square has:

   first digits: 24 and one fewer
  9 than 9's (in the number)      2 4 9 9
next digits: same number of 0's
  as 9's (in the number)                  0 0 0
last digit: 1                                    1

3. So 4999 × 4999 = 24,990,001.

   See the pattern?

1. If the number to be squared is 499999:
2. The square has:

   first digits: 24 and one
  fewer 9 than 9's          2 4 9 9 9 9
next digits: same number
  of 0's as 9's                        0 0 0 0 0
last digit: 1                                    1

3. So 499999 × 499999 = 249,999,000,001.