Welcome! Take three deep breaths, each time expel the air through your nose with force. Now you are ready for this adventure. Let us dive right in and reflect on the premise of this blog post.
I, personally, dislike the word “solve”. What does it even mean? And that is already the second time I have used that word. The word solve implies all kinds of nonsense, such as completion or the existence of a solution. Let us recast it as: new insights, positive reframing or simply “ah-ha!”. These “problems” can be anything too: emotional ones (external or internal), scientific or research ones, artistic ones, writing ones. If you feel like it, just call it a problem.
Whoever is writing this blog post, he certainly does talk a lot … We should really get going or we will run out of time. You know what, let us start again. Welcome to:
Ten Simple Rules for Sol.. (cough) Ten simple Rules For Saying “ah-ha!” more often
The ideas in this blogpost are not original. Many more people have written on the so-called pedagogy of problem solving. There is no intention or desire to be exhaustive or evenly represent the ideas in the literature. I chose to highlight ideas I find curious, interesting and useful. Some other people and places you can find these ideas:
Polya – How To Solve It
Zeitz – The Art And Craft Of Problem Solving
Engel – Problem-Solving Strategies
Newport – Deep Work
Newport – So Good They Can’t Ignore You
Steinhardt – Research As A Stochastic Decision Process
Darwin – Life and Letters, An Autobiographical Chapter
Arthur Conan Doyle – Various works
Rule 1: Know That You Are Stuck
To solve a problem, we need to know that we have a problem. This may be a frivolous point but it is not always obvious that we are in the jaws of a problem. For example, I rewrote that last sentence about twenty times before I realised I was stuck on what I wanted to say. Key signs that you are stuck:
- I do not know how to proceed.
- I haven’t got a clue.
- Why is this not working?
However, it can be more subtle than that. Repeatedly trying over and over again without progress is also being stuck. One scenario of which I am guilty is the following: I want to solve a problem but I am doing other things instead. I refer to this last point as being passively stuck. To know that we are stuck, we have to realise a solution is not immediate. To get better at this, I recommend saying out loud “I am stuck” whenever it is relevant. Others might prefer to write it down on a post-it note or in a lab book, record it on their dictaphone or shout it at their therapist. So to summarise: identify you have a problem, be aware the solution is not immediate, state that you are stuck.
Rule 2: Understand The Problem
Often a problem is hard because we are unsure what we are being asked, or asking ourselves, to do. Poorly defined problems rarely lead to quick “ah-has!” . Spending time to understand the problem is a neglected part of the problem solving process, despite many nostalgic memories of “I don’t even understand the problem”. Typically, there are many strategies for demystification. One way is to rephrase the problem in several manners, each time using simpler and simpler language. Another is drawing a picture or diagram. I often ask myself: Do I have enough information to move forward? Finally, refactoring a problem can be useful; that is, changing the structure of the problem to make it more understandable.
Rule 3: Picture The Solution
Even if you found the solution, would you know it? Suppose someone asked you what seven multiplied by eight was and you scratched your head. They then proceed to say fifty-six. Would you then go “yeah of course” or “ah-ha!”. Drawing a picture of the solution can be surprisingly insightful. It does not matter if the solution ends up being different to what you draw – just ask Da Vinci – it can still be useful. Picturing the solution helps us know when we have found the treasure. It also facilitates backward problem solving. For example, suppose I did not know the answer to seven multiplied by eight. I picture the answer as a number, say, 100. I do know that 100 is 10 multiplied by 10. I have already learnt several things. I am looking for a number and that number is probably less than 100.
Rule 4: Simplifying And Specialising
Few recipe books say: “well just now make the cake”. There is usually a list of ingredients and a list of maneuvers. Ingredients without any instructions or instructions without any ingredients would be infuriating. It is unlikely that we would quickly arrive at a cake by randomly using the ingredients. Breaking a problem down into smaller steps can be illuminating and solving each sub-problem one at time is a more manageable task. You might want to make a 7 tiered wedding cake. Instead of embarking on this challenge, why not attempt a 1 layered victoria sponge first? Considering a specific scenario of your problem can expose its constituent parts. Mathematicians will often call these special cases or basis cases.
Rule 5: Think Like A Scientist
Despite not all problems being scientific ones, thinking like a scientist can be rewarding. As a (good) scientist you generate hypotheses about your problem. For each one, rank them by how plausible they might be. Example: (1) I found a specific edge case that broke c++ (2) I spelt lnegth incorrectly. Another example (1) My sister is mad specifically at me (2) My sister is mad at something and I just happen to be present. I let you decide which are more likely. Begin with the simplest avenues and eliminate those that are unreasonable or unlikely. Finally, have a plan and execute it. Know what you are going to do and have a reason for doing it.
Rule 6: Guess And Check
We can simply be paralysed by not knowing which direction to take. First, return to rule 5 before trying anything else. If we are still stuck, then let us just try something at least. Someone had to be the first person to mix flour and water, or eggs and sugar together. Trying something we have little confidence in can raise uncomfortable feelings. Be prepared to feel uncomfortable or silly – a bad answer can still be useful. We may miss the mark but then we can orient ourselves around this. Once you have taken a guess remember to check if we have made any progress – have we learnt anything new about the problem?
Rule 7: Auxiliary Problems
For somewhat challenging tasks it may be that we have made little progress thus far. In this scenario, we may consider solving a different problem. Here, we look for analogous or related problems which might have already been solved by someone else or the path might be more amenable to the previous rules. On the other hand, we can vary the current problem. We can ask ourselves if a variation of the problem is easier. For example, if we are practising a piano piece, perhaps we can try it really slowly at first.
Rule 8: Go For A Walk
Our environment influences how we think and perceive. Many humans have argued they generate their best ideas in specific places – the shower comes to mind. Others recommend a long walk. If you do go for a walk notice how hard it is to hold abstract ideas. Allow yourself to be distracted and notice things that are both near, as well as far. Find out what works for you by trying different perturbation to your environment. Find a Watson and take them on a walk – explain to them your problem. This is also an opportunity to upset your rhythms. Even something as benign as writing with a pencil rather than a pen can shift your thinking.
Rule 9: Meditate
Problem solving can lead to unwanted or distracting emotions. Frustration, anger and helplessness are just a few. Lack of mindful clarity or attention can hold you back from focusing on your problem. Try to reframe emotions. You are likely angry because you expected to solve the problem by this point. Appreciating that this means you underestimated the problem and it was more substantial that you first thought – and hence more exciting – might refocus the mind. Notice your feelings and be mindful of your thought processes. How did you arrive? I recommend picking up a book on meditation or mindfulness. Here is a quick exercise for beginners: Three deep breaths – forcefully pushing the air out through the nose. Now, breath in, for a count of four, through the nose. Notice the feeling of the breath on the inside of the nostrils and the pressure on the outside of the nostrils. Be aware of the movement of the diaphragm. Hold the breath for a count of four and breath out for a count of six. Notice this time the pressure on the diaphragm and on the inside of the nostrils. During the whole process try to be aware of the continuity of the breath and clarify the end of a breath in and the end start of a breath out. Notice distractions and send them away.
Rule 10: Reflection
Now it is time to reflect and review the process. Where did you have your “ah-ha!” moments? Did they follow from anything in particular? Did anything trigger an “ah-ha!” moment? Which strategies did and do not work for you? When did you fail to move forward? At what points were you set back? Be critical with your review process, but do not discount something simply because it did not work this time. The goal is to get to a point where you can predict the strategy for future problems.
So those were my 10 simple rules. Now you have the recipe, it is time to write your own.
By Oliver Crook, feedback warmly welcomed at oliver.crook at stats.ox.ac.uk.
