0. Remember that problems are soluble.
As you embark on solving a problem, first keep in mind that interesting problems have solutions waiting to be discovered. Without this foundational understanding, you may find yourself mired in inaction before you even begin. Be receptive to the possibilities of what a solution could look like. Allowing the thought that there may be no solution to take hold can limit your ability to perceive and connect valuable information as you move forward. Your preconceptions act as a lens, shaping your perspective and determining which solutions you create.
1. Ask the meta-question: why is this even a problem?
Is it truly a problem and if so, why? What is this problem obstructing us from achieving? By answering these questions, we establish a contextual frame and a reason to solve the problem. Getting clear on the "why" behind the issue will not only jumpstart our creativity and resourcefulness but also determine our level of motivation to find a solution.
2. Question your basic assumptions.
What unexamined axioms are governing your thinking? Are they really true? Are your default constraints merely arbitrary or actually necessary?
3. Invert the problem.
In some cases, inverting the problem might help you solve it. For the mathematically minded, instead of finding p(X=x), it might be easier to find p(X=x') first. For the philosophically minded, this could be similar to taking a via negativa approach.
4. Shrink it down and picture the lower-bound edge case.
Imagine tackling the problem for the smallest possible input, say n=1. If you only had to solve the problem once, for one person, or for one iteration, how would it work? How would scaling affect your solution?
5. Scale it up and picture the upper-bound edge case.
Now take the opposite approach and imagine the problem at its largest possible scale, perhaps n = 1 million or even 1 billion. How would the solution look different on this massive scale? What would be the ultimate limits to your approach?
6. Make analogies.
Analogies can provide fresh perspectives, allowing you to tap into more creative solutions. First, create an analogy of the problem. By creating a different frame of reference, you can explore the problem space from new vantage points. Then, create an analogy of the solution. Testing the variability of your solution through analogy can help you identify weaknesses in your explanation. This is based on the Deutschian notion that good explanations are hard to vary.
7. Draw it out.
In its essence, writing is simply the physical manifestation of thought. But we must not forget that the origins of writing and language are firmly grounded in the symbolic representation of ideas. Long before the advent of alphabets, written language and mathematical notation, our ancestors used pictorial symbols to convey meaning. Drawing diagrams and schematics is a direct extension of this ancient tradition and can deepen our understanding in ways that writing alone cannot. Visual representation has the power to illuminate and captivate in ways that are often lost in the written word.
8. Over to you
What are your principles for solving problems?
Great list!