Without using a calculator, compute $1003^2-997^2-1001^2+999^2$.
I notice that the numbers are symmetrically arranged around 1000, with differences of -3, -1, +1, and +3. This makes me think of using the difference of squares formula, which says that $a^2-b^2=(a+b)(a-b)$. I can group the terms in the expression as follows: $(1003^2-997^2)-(1001^2-999^2)$. Then I can apply the difference of squares formula to each group: $(1003+997)(1003-997)-(1001+999)(1001-999)$. This simplifies to $(2000)(6)-(2000)(2)$. Now I can factor out 2000 from both terms and get $2000(6-2)$. This is equal to $2000(4)$, which is just $8000$. 