The Vast Vast World of Problem-Solving — 1
Last question of the test. You’re stuck on this one problem and you just don’t know how to start. We’ve all been there before: not knowing how to even begin to approach a problem. In that stressful situation, many might just try things randomly to see if one works. When the stakes are high, however, you should be doing the exact opposite to problem-solve in an organised manner. With that goal in mind, welcome to the first part of this series on problem-solving.
When I started to research more about problem-solving, I wanted to analyse problems so I could best approach different ones and set myself up for success. What I learned was that there were more types of problems than I ever imagined and they weren’t just distinguished by fields. The very same problem that a medical researcher might face can also apply to an art critique and it can also be solved in similar ways.
For instance, consider a scientist establishing new applications for a class of drugs and a critic examining the symbolism behind artwork. They both start by establishing their goal and then examining the existing information about that problem. The scientist might explore clinical trials about current uses of the drug, whereas the artist might look at the message behind other works of art in the same genre. Initially, both of these problems require general data collection before focusing in on the specific issue at hand. As I like to say, “you have to get in the same box as everyone else before you can think outside it.”
Then, both the scientist and the artist would use these general details to focus in on the problem at hand. The scientist would identify extensions of existing uses of the drug and the artist would identify symbolism in the art pertinent to its genre. After drawing their conclusions, they would evaluate limitations for potential extensions of their work. In this process, the problem-solving:
- Began by recognising the problem;
- Diverged to gather general details;
- Converged back to the specific problem;
- And then continued testing to verify the conclusion.
Keep this underlying process in mind as I get into some specific strategies I found useful for these steps.
Not Specific Enough? But it’s Specifically Abstract…
Abstraction is used to diverge from a specific problem to the core features of that type of problem in general. The details of the problem are diluted to their simplest forms to apply the concepts behind the specific problem to anything like it.
To illustrate, imagine a company manufacturing an extremely durable tennis ball made of a proprietary material. Its specific problem would be to make a ball made out of that material which would last through as many tennis games as possible. This problem could be abstracted, however, to a ball of any material for any sport that needs to withstand many impacts over its life.
The advantage of abstraction, in particular, is understanding the fundamental factors that influence a problem and simplifying the challenges to overcome. Moreover, it allows for comparison between a new solution for a specific problem and existing solutions for similar problems. For instance, the tennis ball manufacturer could research how baseball manufacturers let their balls withstand many impacts. This process of comparing a new problem to an existing one is where reduction and analogy come in.
Copy/Pasting Solutions to Infinity and Beyond… with Reduction!
Reduction is used primarily in computer science, where a solution of one problem is used to solve a related problem. From the previous example, strategies used to make baseballs more durable could also be used to make tennis balls more durable. A key point to note is that reduction means that solving a new problem (tennis ball durability) is always harder than solving the first (baseball durability). Intuitively, this is because the solution from one problem always has to be modified at least slightly for the second similar problem.
Perhaps, the tennis ball manufacturer discovers that baseballs are more durable with multiple, separated layers. It could then use that solution to solve its own problem, but it would still have to make certain changes. To demonstrate, the company could make tennis balls with multiple layers, but it would still have to account for the balls being hit with racquets instead of bats. In this way, it uses the solution for making durable baseballs to make tennis balls durable, but it still has to change that other solution in some ways.
So let’s Pretend my Analogy is Accurate…
Similarly, creating analogies is another method of using existing information to find a solution to a new problem. To more easily understand the problem at hand, you compare it to another simpler problem through an analogy. The key detail here is that the other problem is simpler. Analogies aren’t like reduction where the solution for one problem is slightly modified to solve another. Since the analogy makes a comparison to a simpler problem, some accuracy is lost and the solution to that simpler problem isn’t fully usable for the first.
To switch up the examples, say a teacher challenges students to describe how a tree makes energy. The problem for the students (describing a tree’s energy production) can be made simpler by using an analogy of the tree being like a coal plant. The students could describe that sunlight is like coal and chloroplasts are like steam turbines that create glucose (the tree’s electricity). In this, the comparison isn’t entirely accurate, yet it still helps the students accomplish their goal by comparing a harder problem to a simpler one.
You do not, however, need to look to other problems to simplify the one at hand. Methods like divide and conquer let you simplify the problem you have itself.
Why Listen to the Romans when you have Scientists?
Divide and conquer is a fairly well-known saying with confusing roots. It is earliest traced to the Roman era as a military/political strategy. Its initial meaning, however, was to divide yourself and conquer the enemy, instead of dividing the enemies to conquer them. Nevertheless, the principle behind divide and conquer problem-solving techniques today is to break down a difficult problem into smaller, simpler problems to solve, much like dividing a powerful enemy into weaker groups. This technique is often used in computer science as a way to decrease the complexity of demanding algorithms.
To enumerate, imagine that someone in New York is trying to find the fastest way to reach the top of Mount Everest for their ideal vacation destination (I’m not judging…). The initial problem is hard to approach, as someone could consider a huge amount of possible routes to take on their vacation. This problem, however, could be broken down in simpler phases. Primarily, the person might find the fastest flight to get from New York to Kathmandu, Nepal. Next, they could consider the quickest means of transportation to get to the bottom of Mount Everest 100 kilometres away. Finally, they would break down how to most quickly get to south or north base camp and so on. As can be seen, solving these smaller problems is easier and less daunting than solving the original problem without breaking it down. Herein lies the advantage of the divide and conquer method; big problems may seem impossible to solve, but the big problem can be broken down into smaller problems until one of those is solvable.
- The best way to solve a problem isn’t whatever comes to your head first;
- Multiple fields have similar problems and techniques to solve them;
- You have to understand a topic in general before solving a specific problem;
- Existing solutions can be used to solve new problems;
- And it is easier to solve multiple smaller problems than one big problem.
Moreover, a difficult problem can be broken down into underlying causes as well as underlying problems. This is the focus of the next part of this series on root cause analysis, one of my favourite techniques in problem-solving.