Suppose *a* = 2485144657 and *b* = 7751389993.

- What is the last digit of
*a***b*?
- What is the median digit of
*a***b*?

In both questions it is *conceptually* necessary to compute *a***b*, but not *logically* necessary. Both are a question about *a***b*, so computing the product is conceptually necessary. But there is no logical necessity to actually compute *a***b* in order to answer a question about it.

In the first question, there’s an obvious short cut: multiply the last two digits and see that the last digit of the product must be 1.

In the second question, it is conceivable that there is some way to find the median digit that is less work than computing *a***b* first, though I don’t see how. Conceptually, you need to find the digits that make up *a***b*, sort them, and select the one in the middle. But it is conceivable, for example, that there is some way to find the digits of *a***b* that is less work than finding them in the right order, i.e. computing *a***b*.

I brought up the example above to use it as a metaphor.

In your work, how can you tell whether a problem is more like the first question or the second? Are you presuming you have to do something that you don’t? Are you assuming something is logically necessary when it is only conceptually necessary?

When I’m stuck on a problem, I often ask myself whether I *really* need to do what I *assume* I need to do. Sometimes that helps me get unstuck.

**Related post**: Maybe you only need it because you have it