Problem solving

Generic strategies

Brooma, October 2019

Definition

Problem solving is the act of defining a problem; determining the cause of the problem; identifying, prioritizing, and selecting alternatives for a solution; and implementing a solution.

American Society of Quality

What is problem solving?

Problem solving is a process of defining a problem; determining the cause of the problem; identifying, prioritizing, and selecting alternatives for a solution; and implementing a solution.

If a process doesn't have all these stages, then this process is not a an process of problem solving. Let's go though all parts of this definition.

You simply cannot solve a problem if you didn't define a problem, so you just don't have it.
If you don't determine the cause of the problem, then you just don't solve the problem. You do something else.
If don't identify possible solutions, then you just can't go through the solution of the problem. If you don't prioritize a solution, you can't select a solution, and if you don't select a solution, then you just can't start resolving the problem. But sometimes you can do even all these steps by saying: "Let's just use this solution". It this case this possible solution can be the solution and may be not a solution at all.
If you don't implement a solution, the you will not solve the problem. Probably you just research the problem and approaches.

Bites of knowledge

problem-solving strategies
problem-solving methods
metacognitive knowledge

For the purpose of simplicity of understanding this report is presented as a list of bits of knowledge.

Problem-solving strategies are the steps one can use to solve the problem. E.g. proof by contradiction. Strategies can be generic is such a way that it can be relevant for most problems, and can be specific, e.g. can describe only approach to solve programming problems on arrays.

Problem-solving methods are systematic approaches to solve problems. E.g. protocol of urgent bug resolution.

Metacognitive knowledge is what individuals know about themselves as cognitive processors.

This report is mostly about strategies. Also it contains some approaches to improve problem solving in the long run. These approaches are based on highly trivial metacognitive knowledge.

Let's suppose that there are two most fundamental questions that you can ask yourself about the problem immediately:

Should I solve the problem at all?
Is it solvable?

The first one is usually quite simple, but the second one deserves a separate report. But because usually we solve solvable problems, let's start talking about strategies of problem solving.

Strategy 1

Find a ready strategy

Yes, just find a ready strategy!

Strategy 2

Division

Just divide the problem to subproblems

It makes sense no matter if your problem is to understand a problem, or to find a solution.

Strategy 3

The most general algorithm

Understand the problem
- Understand what we have
- Understand what we want to achieve
Find the path from the given state to the desired state
- Find a division to understandable and solvable subproblems.
  - Apply this algorithm to subproblems.
Follow the path

Strategy 4

Enumeration

Enumerate all the possible variations of solutions or proofs (brute-force search).

Strategy 5

Patterns

Look for patterns

Flavor 1

First principles

Flavor 2

Abstraction

Abstraction makes the concept more vague.

The class concept is vague:

a publication
- a newspaper

Flavor 3

Algorithmic paradigm

Classification of algorithms by design paradigm

Flavor 4

Design patterns

Inventor's paradox

"Instead of solving a specific type of problem, which would seem intuitively easier, it can be easier to solve a more general problem, which covers the specifics of the sought-after solution."

AOSA

Strategy 6

Constraints

Not all solutions are equally useful.

Strategy 7

Incremental solving

Flavor 1

Offer and refute

Repeat:

Offer a theory
Try to refute it
After each satisfaction of a previous condition, you can try to find a simpler theory.

Flavor 2

Declare and define

Declare your function
Define it
- Write the definition using pseudocode
- Replace each term to a variable use or a function call
- Declare each variable and function
- Define each variable and function

Flavor 3

Test-driven development

The dangerous one

Trial and error

Testing possible solutions until the right one is found. It's an inherent empiric method of human thinking.

Strategy 8

Take into account the reality

Strategy 9

Reduction

Transform the problem to a solved one.

Approach 1

Research

You can never know enough mathematics.

How To Think Real Good

It's quite understandable and widespread that those who know more solve problems better.

In order to solve specific problems you could think like someone who already solved that kind of problems in their scientific work. Can you guess that you could think like they when you don't know their problems and solution? No, you can't. Problems and solutions can be very very unexpected. The best thing you can do is to find areas of knowledge and problems and understand how that problems are solved by professionals, i.e. learn mindsets of that professionals. I mean yes, it doesn't mean exactly that you have to know the math.

For example my friend (how is very productive scientist) say that he solve problems for one hour per day. All the other time he learn new things. And quite frequently when he need to make new research he just say: "Wait a second, I've just read something similar. Let's try this." And it helps.

Problem solving

Generic strategies

Definition

Bites of knowledge

Strategy 1

Find a ready strategy

Strategy 2

Division

Strategy 3

The most general algorithm

Strategy 4

Enumeration

Strategy 5

Patterns

Flavor 1

First principles

Flavor 2

Abstraction

Flavor 3

Algorithmic paradigm

Flavor 4

Design patterns

Inventor's paradox

Strategy 6

Constraints

Strategy 7

Incremental solving

Flavor 1

Offer and refute

Flavor 2

Declare and define

Flavor 3

Test-driven development

The dangerous one

Trial and error

Strategy 8

Take into account the reality

Strategy 9

Reduction

Approach 1

Research

Strategy 10

Creative approaches

Approach 2

Build your problem-solving arsenal

Flavor 1

Self-reflection, yaaay!

Flavor 2

Write your own problem-solving algorithm

Thanks 👏🏼