Squaring numbers in the 400s

1. Choose a number in the 400s (keep the numbers low at first; then progress to larger ones).
2. The first two digits of the square are 16: 1 6 _ _ _ _
3. The next two digits will be 8 times the last 2 digits: _ _ X X _ _
4. The last two places will be the square of the last two digits: _ _ _ _ X X

   Example:

1. If the number to be squared is 407:
2. The first two digits are 16: 1 6 _ _ _ _
3. The next two digits are 8 times the last 2 digits:
   8 x 7 = 56: _ _ 5 6 _ _
4. Square the last digit: 7 x 7 = 49: _ _ _ 4 9
5. So 407 x 407 = 165,649.

For larger numbers reverse the steps:
1. Square the last two digits (keep the carry): _ _ _ _ X X
2. 8 times the last two digits + carry: _ _ X X _ _
3. 16 + carry: X X _ _ _ _

   See the pattern?

1. If the number to be squared is 425:
2. Square the last two digits (keep the carry):
   25 x 25 = 625 (keep 6): _ _ _ _ 2 5
3. 8 times the last two digits + carry:
   8 x 25 = 200; 200 + 6 = 206 (keep 2): _ _ 0 6 _ _
4. 16 + carry: 16 + 2 = 18: 1 8 _ _ _ _
5. So 425 x 425 = 180,625.