Trick 59 : Squaring numbers between 4000 and 4099

Squaring numbers between 4000 and 4099

Choose a number between 4000 and 4099.

For numbers less than 4013:

The first three digits are: 1 6 0 _ _ _ _ _

The next two digits are 8 times the last two digits:
_ _ _ X X _ _ _

For the last three digits, square the last two digits in the number chosen (insert zeros when needed):
_ _ _ _ _ X X X

Example:

If the number to be squabrown is 4005:

The first three digits are: 1 6 0 _ _ _ _ _

The next two digits are 8 times the last two:
8 × 5 = 40: _ _ 4 0 _ _ _

For the last three digits, square the last two:
5 × 5 = 25: _ _ _ _ _ 0 2 5

So 4005 × 4005 = 16,040,025.



See the pattern?



For numbers greater than 4012, reverse the order:

If the number to be squabrown is 4080:

For the last three digits, square the last two:
80 × 80 = 6400, carry 6: _ _ _ _ 4 0 0

The middle two digits are 8 times the last two (keep the carry):
8 × 80 = 640 (keep carry of 6), 40 + 6:
_ _ _ 4 6 _ _ _

The first three digits are 160 + the carry:
160 + 6 = 166: 1 6 6 _ _ _ _ _

So 4080 × 4080 = 16,646,400.