How do we solve problems? How do you teach others on your team to solve problems? Have you consider a spherical cow point of view in solving problems?

In a recent nugget, I talked about complexity and how to stop it from being a “hot potato” that keeps being thrown into your direction. In problem-solving, we seek to reduce complexity as quickly as possible; intelligently, of course. I mentioned the complexity quotient (CQ). A person can raise their CQ through practice with dealing with complexity.

In this nugget, let’s talk about** another aspect of problem-solving: Approximations**.** Approximations you can do in your head or on the back of an envelope**. In other words, how can you estimate an answer without doing detailed calculations on your computer?

I worked as a teaching assistant for undergraduate chemistry students in graduate school.** I was often amazed at students “blindly” writing down the answers they got from their calculators. They didn’t check whether their result made sense.** If the answer was much larger or smaller than even made sense, they often did not stop to notice.

### A Learned Skill: Look At The Big Picture While Having Intricate Details Before You

**Checking whether an answer makes sense is a learned** **skill. It is part of problem-solving. You have to look at the big picture even though you also deal with intricate details.**

When I was a postdoctoral fellow, I learned about a great book by John Harte, “ Consider A Spherical Cow: A course in environmental problem-solving.” The book’s object is to** practice estimating an answer to a question within an order of magnitude**. The book is valuable far beyond the realm of environmental problem-solving. The thought process applies to many different types of problems.

### A Simple Example

Here is a simple example (the approach is from the book; the specific example is not). Let’s say a **friend tells you he heard in the news that the number of cars registered in the U.S. has jumped to 600 million vehicles.** You think, “*Wow, this sounds quite high!*” You wonder: “*Can this number be valid?*“

Of course, you could search for the answer on the internet. But what is the fun in that? Plus, we are talking about solving problems here, and many have no answer on the internet.

**How could we get to the approximate answer through estimating?**

- We have to ask a series of questions and make estimates.
- All estimates will be somewhat wrong. But there are likely some error cancellations.
- Furthermore, we are not going after the exact answer. We want an approximate one to see whether what our friend said makes approximately sense.
- Whatwe are doing it like a litmus test.

### How many cars are registered in the U.S.? Is 600 Million a reasonable number?

- Let’s consider that the U.S. has a population of about 330 million people. Even if you think it is 300 Mill or 350 Mill people, 600 Million cars sound suspicious right away. The friend’s statement means we have nearly two vehicles per person, including kids under 16?
- We could assume that 75% of the population is 16 years of age and older. That is about 250 million people who could potentially drive a car. That is still a significant number but more than a factor two less than what the friend said.
- Further considerations:
- Not everyone who can drive can afford or even wants a car.
- Not all teenagers own a vehicle.
- Some seniors can’t drive anymore.
- Not all families have two cars. Some families may have more than two cars.

- A crude assumption would be that 50% of everyone who could drive owns a car. That would be 124 Mill cars.
- Even if we assume that 75% of people over 16 have cars, the answer (186 Mill) is still far less than 600 Mill.

### The Thought Process Is The Point. How Do We Dissect The Problem?

**The point here is not the math per se. It is not about this simple example either**.

**It is the thought process. How do we dissect the problem**? As it turns out,** dissecting is also about reducing complexity**. We make that cow spherical to think our way through the problem without getting tangled up because the cow has legs, a head, and maybe horns.

## I’m Curious

- How do you problem-solve?
- How do you teach that to others?

P.S.: I appreciate you commenting and sharing this with others. Thank you!