Sean Cheng

How To Solve Hard Problems

February 21, 2021 - 10 min read

I’ve noticed something about the people I’ve most enjoyed working with.

They have a knack for solving hard problems.

They’ll send out emails, get on calls, go door to door if necessary to get stuff done.

Our livelihoods depend on our ability to problem-solve. If you can’t solve problems, you aren’t providing value. If you aren’t providing value, you don’t get paid. And if you don’t get paid? Well, I think you know where it goes from here.

By definition, solving hard problems is hard. That’s why entrepreneurs who create innovative solutions, engineers working on the cutting edge of artificial intelligence, and elite neurosurgeons operating on the most challenging patients are compensated so well. We reward those who provide value to our lives. These individuals are difficult to replace and as a consequence, have more freedom to choose what they work on and who they work with.

While the specifics of how a problem gets solved varies by industry, the core principles remain the same. By mastering the mental models high performers use when they encounter challenging situations, you can take your work to the next level. This essay outlines these strategies.

Think From First Principles

First-principles thinking is a physics-based mental model for problem-solving. Elon Musk used this method to break down challenges he encountered while working on Tesla and SpaceX.

He summarizes the approach like this:

“I think it’s important to reason from first principles rather than by analogy…What that means is you kind of boil things down to the most fundamental truths and say what are we sure is true and then reason up from there.”

First-principles thinking requires you to let go of your intuition and assumptions, and do the intellectual work to make sense of the world from its most basic building blocks. It helps you break down complex problems into manageable chunks.

An example Musk brings up is how he used first principles to design the Tesla car battery packs. Originally, people assumed batteries would always be too expensive to use for electric vehicles. But Musk broke down this assumption by first thinking about what elements each battery consisted of, then doing a cost-analysis of all the raw materials. Raw materials ended up being relatively cheap and he cut costs by manufacturing his own batteries.

One way to establish first principles is with the “why” technique. When you encounter a problem ask yourself why it’s occurring until you get the root cause.

A scenario where you are struggling to put on muscle might play out like this:

  1. I’m struggling to gain muscle. Why?
  2. I’m not in a caloric surplus. Why?
  3. It’s challenging for me to eat a lot during each meal. Why?
  4. I have a small stomach capacity. Why?
  5. Because I do.

At this point, you’ve reached the root cause of having a small stomach capacity and you can begin thinking of ways to sidestep this obstacle. You might consider eating more meals during the day or drinking smoothies to get in extra calories since liquids tend to not be as filling.

🔥 Enjoying reading this? If you'd like to receive exclusive early access to new posts, along with reading recommendations and other insights, consider providing your email below. I don't spam and will be releasing writings on how to be a better leader in the coming months.

Look For Structural Similarities

In situations where first principles leaves you stuck, reasoning by analogy can be a viable alternative. The trick is when you look for analogies, you must overlook surface-level details and seek structural similarities.

Here is a short thought experiment proposed by cognitive psychologist Karl Duncker demonstrating this idea:

Suppose you are a doctor faced with a patient who has a malignant tumor in his stomach. It is impossible to operate on the patient, but unless the tumor is destroyed the patient will die.

There is a kind of ray that can be used to destroy the tumor. If the rays reach the tumor all at once at a sufficiently high intensity, the tumor will be destroyed. Unfortunately, at this intensity, the healthy tissue that the rays pass through on the way to the tumor will also be destroyed.

At lower intensities, the rays are harmless to healthy tissue, but they will not affect the tumor, either. What type of procedure might be used to destroy the tumor with the rays and at the same time avoid destroying the healthy tissue?

Stumped? Here’s a story.

A small country was ruled from a strong fortress by a dictator. The fortress was situated in the middle of the country, surrounded by farms and villages. Many roads led to the fortress through the countryside. A rebel general vowed to capture the fortress.

The general knew that an attack by his entire army would capture the fortress. He gathered his army at the head of one of the roads, ready to launch a full-scale direct attack. However, the general then learned that the dictator had planted mines on each of the roads. The mines were set so that small bodies of men could pass over them safely since the dictator needed to move his troops and workers to and from the fortress.

However, any large force would detonate the mines. It, therefore, seemed impossible to capture the fortress. However, the general devised a simple plan. He divided his armies into small groups and dispatched each group to the head of a different road.

When all was ready, he gave the signal and each group marched down a different road. Each group continued down its road so that the entire army arrived together at the fortress at the same time. In this way, the general captured the fortress and overthrew the dictator.

Did that help?

In case you still didn’t solve the problem, the answer is the doctor should use multiple low-intensity rays to attack the tumor from all sides. This prevents the rays from destroying the healthy tissue while also allowing the rays to be intense enough to destroy the tumor.

The analogy of the general and dictator follows the same logic. By dispatching small groups of soldiers to attack the fortress from all sides (i.e. the low-intensity rays), they were able to capture it without setting off the mines (i.e destroying the tumor without killing healthy tissue).

According to a 1980s research study conducted using these exact same prompts, about 10% of people solve the initial problem, but when presented with the fortress story, 30% do. Reasoning by analogy works because it allows you to access new information and perspectives beyond the domain you’re focused on.

Not every analogy will be effective. You need to find structurally similar analogies because finding analogies similar only at the surface level doesn’t provide you with new information to work with. You’re getting more of the same thing.

For example, if you’re solving a business problem in the artificial intelligence space, looking at what other AI companies are doing won’t help you arrive at an innovative solution. Most likely, they’ve tried the same tactics as you. But looking at how a fast food restaurant solves a structurally similar problem might lead to a breakthrough, even if the surface level details seem incomparable.

The best way to identify structurally similar analogies is simply being mindful about seeking them out. University of Sydney Business professor Dan Lovallo along with collaborator Ferdinand Dubin found that simply prompting business students with analogies helped them generate more ideas. Reading widely and exploring tangential interests can also help you identify connections between seemingly disparate fields.


Most people overvalue innate intelligence. Oftentimes, what we perceive as intelligence is merely a well-read individual with an above-average work ethic.

The vast majority of situations don’t require a genius to figure them out— you only need to hustle.

Shooting more shots and experimenting with different approaches will result in a higher probability of success. Quantity and quality aren’t diametrically opposed. Quantity leads to quality.

Problems like low sales are examples of this. You might have the best sales pitch, but if you don’t put in the work to talk to hundreds of leads and iterate based on feedback, don’t be surprised if you don’t get the results you want. The “if you build it, they will come” mentality rarely applies to most things.

What I’ve noticed from my own experiences and conversations with friends is trying to have the perfect solution in place before taking action is the most common barrier to finding a way through tough problems.

An example that comes to mind is when I had to source survey responses for my work. I spent the majority of my time trying to perfect the messaging behind my emails to elicit a high number of responses. I thought if I wrote the perfect email then I would be set. Imagine my shock when I garnered a measly 16% open rate on my first round of emails.

Undeterred, I sent out several more rounds of emails and refined the messaging after each batch. By the end, I was getting a 46% open rate.

Hustling is what helps you quickly validate whether or not your “good” ideas were any good, to begin with. In the case of my emails, it’s what allowed me to improve how I framed the messages. Spending too much time intellectualizing doesn’t bring results.

Focused vs. Diffused

We have two modes of thinking. Focused and diffused.

Focused thinking is active and analytical. You’re fixated on the task at hand, gathering data and trying to build understanding in your brain. This step is crucial when you are learning a new concept and are acquiring the foundational blocks of knowledge.

Diffused thinking occurs when you relax and let your mind wander from what you’re working on. This helps you use the parts of your brain not activated when you were focused.

Focused mode must be alternated with diffuse mode when we get stuck on tough problems because of the Einstellung effect. Barbara Oakley, the author of A Mind For Numbers, describes it like this:

“Einstellung effect (pronounced EYE-nshtellung). In this phenomenon, an idea you already have in mind, or your simple initial thought, prevents a better idea or solution from being found.”

The Einstellung effect primes our brain to use the knowledge we think is relevant to solve the problem at hand. But oftentimes, this knowledge can block other helpful details. By relaxing into diffuse mode, your brain can form the seemingly random connections that are the beginnings of creative solutions.

Activating your diffuse mode can be as easy as taking breaks between work sessions to meditate, going to the gym, taking a shower, or listening to music without words. Another method recommended by Oakley is focusing hard on a problem before going to sleep. While you’re sleeping, your brain can find the diffuse connections and when you wake up you may have a novel solution.

This cycle between focused and diffused can and should be repeated multiple times when you’re tackling a tough challenge.

Books I drew heavy inspiration from for this essay:

Subscribe for reading recommendations, exclusive insights, and updates on future posts like this: