### How do we do arithmetic?

If one is asked to multiply 499 and 501 many will do something like the following:
Pattern: (n+1)*(n–1) = n^{2} – 1 thus 499 * 501 = 500^{2} – 1 = 5^{2} * 100^{2} – 1 = 25 * 10000 – 1 = 249999
These are steps that fit in the head more easily than the tedious multiplication rules first taught in school.
How do we notice that 499 and 501 are adjacent to the convenient number 500 and how do we know also that 500 is convenient?
We must realize both of these at once because only the combination of facts is useful.

Tracing the Spark of Creative Problem-Solving

The Ineffability of Insight