On the Art of Problem Solving

What do we mean when we state, “I have a problem!”? That’s one of those sneaky questions which we assume we know the answer to until we actually start answering it. Here are a couple of common responses:

  • When something is broken;
  • When something doesn’t work;
  • When things aren’t going the way I want them to.

With these in mind, consider the following situation:

You walk into someone’s office and they complain to you, “Something’s wrong with my PC, it isn’t working!” You look over and immediately notice that the computer isn’t plugged in.

Question for you … what’s the problem facing you? Is it …

  • The PC isn’t plugged in?
  • The user didn’t notice the computer wasn’t plugged in?
  • The user didn’t know enough to check to see if it was plugged in?

That’s a pretty standard list, but we don’t have to stop there, we could continue …

Is the problem that:

  • PCs aren’t smart enough to plug themselves in?
  • PCs need to be plugged in?
  • Plugs are too easily unplugged?
  • Power outlets are on the wall and not on the desk where we need them?

Each of these problem statements leads to entirely differently solutions, ranging from merely bending down and replugging the computer into the wall socket to a product revolution which will make you the next Bill Gates.

So, back to the original question, what do we mean when we state “I have a problem!”?

It doesn’t automatically mean something is ‘broken’; the unplugged PC above isn’t broken. There isn’t anything wrong with the PC. It’s obeying all the laws of physics. So … what’s going on? The explanation “When things aren’t going the way I want them to” is not so much “more accurate” as it is “more useful” than the notion that “something needs fixing”.

Gerald Weinberg and Donald Gause in their book “Are Your Lights On?” offer the following definition of a “Problem”:

A problem exists,
when there is a difference between …
What we perceive and,
What we desire.

I’ve found this definition incredibly useful as the base foundation upon which to build a problem solving methodology. Notice, it makes no reference to the outside world, all the problems exist somewhere in the gap between desire and perception.

The simple example of the unplugged computer points to the first important step in problem solving. We must agree from the start what exactly it is we intend to work on. We must gain consensus on the problem definition before we do anything else.

To become more profitable … do we increase profit margins? Reduce costs? Or increase sales?

To ‘fix’ this machine, do we … learn how to fix the machine? Find the part that is preventing it from doing what we want it to? Stop the thing that’s causing it to break in the first place? Find the right person and pay them to fix it? Replace it? Or change our way of doing things so we don’t need to fix it? Etc. etc.

One of most common bits of advice in problem solving is, “Never latch onto the first solution you find!”… that’s great advice. Equally great however is the advice, “Never latch onto the first problem description you come across!”

A good problem definition is almost, not always, equivalent to the problem’s solution. The observation that the PC is unplugged screams at you to plug it in. While that solves the immediate problem, it leaves open the other problem of why wasn’t the user able to figure that out for themselves?

This leads directly to a specific PS tip. Never, under any circumstances, automatically accept a problem definition from the person bringing the problem to your attention. The reason is simple enough, if they really knew what the problem was, then they wouldn’t have a problem. That’s not always true, but it’s true often enough to help the self ordained problem solver avoid a lot of grief.

Next PS tip for the PS definition stage. When we’re trying to get agreement on a PS definition we tend to ask questions which seek an affirmative answer. We do this for at least two reasons:

  1. To demonstrate our understanding of the problem (ie. To look good/smart)
  2. To get ‘agreement’ – after all, that’s what we’re seeking!

What is also useful, perhaps even more useful, is to ask questions which deliberately seek a negative response. Ie. No that’s NOT what I meant!

Why would you do this? To erect concrete, firm boundaries on the problem definition. A firm “No!” is worth at least a dozen casual nods of affirmation. This is partly because people tend to ‘agree easily’, sometimes too easily. They nod their heads in agreement without really listening to what you were saying to them.

This technique is also useful for testing assumptions about a problem. If one of your goals is to “Think outside the Box” then one way of finding out where the walls of the box are, is by throwing yourself against your assumptions and taking note of where you get bruised and bounce back stunned.

Since we’ve wandered into the topic of creativity by way of the “Think outside the box” path, it’s worthwhile spending a little bit of time pulling down a creativity myth.

Being creative is not about striving to think differently, it’s much more about listening to yourself and paying attention to the little voices in your head — Okay, that obviously needs some explanation – I’m not suggesting you hear voices (actually I’m suggesting we all hear voices, and that’s a good thing.)

In one of my workshops I give a specific task to the audience. As they’re working on the assignment I wander around the room and listen in on their mumbling conversations. One of the phrases I hear very often is, “This is just like ‘tic-tac-toe’!” Once the exercise is over, I ask for the solutions. Typically nobody has solved the problem. I ask them if they compared the exercise to ‘tic-tac-toe’? Most of them respond yes! I ask them if they attempted to use that information? None of them respond yes.

Basically, when they noticed the similarity to ‘tic-tac-toe’, and even voiced it out aloud, it was their brain screaming for attention and being ignored. Once I tell them to USE the knowledge, they quickly solve the problem.

Lesson? When you notice a ‘pattern’/’similarity’/’metaphor’ use it!

What’s a pattern?

When you see a problem for the first time? That’s an incident… solve the problem.Move on.

When you see a problem for the second time? That’s a coincidence … solve the problem. Make a note of it, move on.

When you see a problem for the third time? That’s a pattern …. Solve the problem. Then find the source of the problem and solve THAT problem. Move on.

Another example of the art of just letting creativity ‘happen’? All of us have something in common. All day long we strive to be creative, and often we fail … yet when we go home, relax, and sleep – we dream the most fantastic things. When we’re trying least, our creativity is unleashed. The challenge is to remove all the inhibitors to creative thought during the day and allow our minds to see patterns and new connections. Creativity is natural, something we don’t have to ‘work at’, we just have to pay attention to our thoughts while problem solving.

Where are we so far? We’ve spoken about what problems are, a bit about problem definitions, and touched on creativity. Now it’s time to get to some hard nosed technical stuff. Specifically ‘What do you know about the problem/situation’, more precisely, “Data Collection”.

Once you’ve agreed on the problem definition, the next step is to focus on “what do you know?” This can get a little bit confusing, as there is a fuzzy boundary between ‘data collection’ and ‘data analysis’. Or at least fuzzy for me in the sense that a lot of the time I can’t see a clear boundary.

The best way is to talk about this phase of problem solving … is to solve a problem. So here’s a simple one.

Arrange the nine numbers from 1-9 into a 3×3 grid, such that each row of 3 numbers, each column of 3 numbers, and each diagonal of 3 numbers all add up to the same number. (p.s. I know that not everyone ‘likes’ math problems, but they often serve as the best puzzles to work with because they are well defined. Even if you hate these puzzles, play along, there’s something here for you, I promise.)

One way to determine “what do we know” is to restate the problem in a slightly different manner. In real life we’d be able to ask the problem giver if our new ‘representation’ is accurate, in this case we’ll have to use our own judgment.

Here’s another way of posing the problem:

In the diagram below, the letters A-I represent the numbers 1-9.
We need to identify the numerical values of A-I


And …

Rows Diagonals Columns

So far, all we’ve done is restate the problem. Now we start the data gathering process…. “What do we know?”

We know that the numbers 1-9 total to 45… (Simple arithmetic)

We now that each of the three triplets (A,B,C), (D,E,F), (G,H,I) total to the same number… X… therefore 3X=45… therefore X=15

Question? (Here’s that fuzzy line) is X=15 ‘data collection’ or ‘data analysis’? Does it matter?

Next step? We’re looking for numerical triplets that add up to 15. How many exist? We need at least eight triplets. What are they?

1+5+9 = 15 our first possible triplet.
1+6+8 = 15 our 2nd possible triplet.
2+4+9 = 15 our 3rd possible triplet
2+5+8 = 15 our 4th possible triplet
2+6+7 = 15 our 5th possible triplet
3+4+8 = 15 our 6th possible triplet
3+5+7 = 15 our 7th possible triplet
4+5+6 = 15 our 8th possible triplet

I’ll leave it as an ‘exercise for the reader’ to prove/demonstrate that I’ve covered all the possible triplets. Remember, no duplicate digits are allowed in the triplet, and the triplet must sum to 15. That means that (5,5,5) isn’t allowed… and there is no triplet that contains a 9 and a 6…

Our task is NOW to place these 8 triplets into the diagram below. (notice how problems naturally break into ‘pieces’… Question? When you put a jigsaw puzzle together… what are the ‘steps’? What do you always do first? What do you do next? And then?)


Is there ‘more’ information available to us?

Look at the corners A,C,G and I … they are each part of three triplets: a row, a column, and a diagonal.

The center (E) is part of four triplets: a row, a column and BOTH diagonals…

And the ‘center sides’ D,B,F,H are part of only two triplets… A column and a row.

Question… is there a number (represented by E) which is contained in 4 of the triplets we found? (a row, a column and the two diagonals) There’s ONLY one number that meets this condition. The number 5! So we have this… E=5

D 5 F

The next step? What are the two triplets that contain the number 5 and contain the ‘center sides (B&H) and (D&F)? ie? D & F and B & H must each be only a part of 2 triplets.

The triplets (3,5,7) & (1,5,9) meet these conditions… so we have…

A 1 C
3 5 7
G 9 I

Now look at the letter ‘A’… Are there three triplets (A,1,C), (A,3,G) and (A,5,I) that allow us to complete the puzzle?

Here are the triplets that meet that condition(8,1,6), (8,5,2), (8,3,4) — giving us…

8 1 6
3 5 7
4 9 2

Viola… (Challenge to the reader… can you think of at least one OTHER approach to solving this problem?)

This might all seem tedious… and it is. The data collection part of PS is ‘grunt work’, there’s nothing very exciting about the collection and categorization of information. In the world outside math problems this data collection is less precise, more ambiguous…. What does the client really want? What are the real deadlines, resources, compromises, constraints? What are the agendas of all the interested parties? Who wants the project to succeed? Who wants it to fail? Why? All of this is important information. It represents the ‘data’ of real world problems.

The data analysis part is also tedious; a bit more intellectually stimulating, but still tedious. What does this bit of information mean? How does it affect the rest of the problem? What are the consequences if we do ‘this’ rather than ‘that’?

Unlike math problems like the one above, not all problems solve ‘themselves’ just by doing the methodical data collection and analysis. A Sudoku problem always falls to the onslaught of the question ‘what do we know?’ and the application of simple rules of logic. In the real world, what’s required is a spark of insight, the moment when all of a sudden we ‘see’ what needs to be done.

How do we get that spark to happen? The truth is, we don’t know. You can give two people the exact same information, the same training, the same experience and one is stumped by the problem and the other person goes ‘Aha!’ as they connect the dots and arrive at the solution.

While there is no way to train someone to find that ‘Aha!’ moment, we can do a lot to make it more likely that we stumble upon it. I’ve collected a few ideas and labeled them ‘Unconscious’, ‘Personal’, and ‘Mechanical’. The intent is to provide some tools that place the spark of “Aha!” under your feet more often as you stumble towards an answer to whatever problem facing you.

Unconscious Tools:
Intended to get you to pay attention to your unconscious.

  1. Don’t push too hard – If you’ve been working for a long time on a problem with no success, then take a break. Do something totally unrelated to the problem. Let your unconscious mind work on it for a while. Watch a movie, read a book, go for a walk, a run or a swim.
  2. Listen to the voices within – If you’ve been noticing something familiar about the problem, then focus on that for while. What’s this problem ‘like’? How is it different from other problems you’ve worked on? How is it similar? Can you make it more similar? Less similar? How? Why not?
  3. Learn to watch yourself – This isn’t easy, but it’s worth practicing. As you solve the problem, what approaches are you always taking? Why? Is it because of habit? Or because you have a reason to approach a problem that way? Can you consciously change your approach?
  4. Write a description of the problem – Don’t JUST think about the problem. Write it down. Diagram it. Write down what you think the problem is, what you think the problem solution would look like. Write down what you know for certain, what you think you know, what you’re assuming. Challenge each point you make. Are you certain you know ‘X’ is true? How do you know that? How could you be wrong?
  5. The Janitor effect – Find someone who knows nothing about your problem. More importantly, find someone who could not possibly solve the problem because they literally don’t know enough about the subject matter … now … explain the problem to them so that they CAN understand it. By forcing yourself to restate the problem in terms they’ll understand, you’ll force yourself to think differently about the problem.

Personal Tools:
Intended to continually sharpen the saw.

  1. Power time – When are you most creative? The most suited to the grind of admin work? When are you best at thinking and best at documenting? Most of us have certain times of the day (or night) when we’re better at different tasks. We can’t do much to change those times, we can take note of them and use to our advantage. I don’t write articles in the afternoon, nor do I waste my creative time in the early morning by doing accounting. I allocate different tasks to different times of the day. Creative tasks when my creative juices are flowing and admin task for when I’m operating on automatic, with not a creative spark in my head. (My accountant and Revenue Canada thank me for this.)
  2. Teams or Solitude – Do you work better on a team or by yourself? There’s no blame attached to a personal preference, it’s just a good idea to know where we shine. If Jim is most creative working by himself in the corner, then it makes little sense to insist that he always work in a team.
  3. Read Everything – Problem solving is mostly about connecting unconnected bits of information. The more information you have at your disposal, the better. You can’t become a good problem solver in any field if you don’t know the terrain. Reading is the #1 way to gather info because you can do it anywhere, anytime. Almost a mundane reminder, but the most powerful tool at your disposal. Read everything. And don’t just read within your field… read everything!
  4. Is credit important to you? – One of the most important questions a Problem Solver needs to ask themselves … Do I want to solve the problem … Or do I want to get credit for solving the problem? The fact is, we can solve far more problems, far more difficult problems, if we don’t care who gets the credit. It’s a question we all have to ask ourselves each time to face a problem. It is far more difficult to get credit AND solve the problem than it is just to solve the problem. I’d recommend anyone take a day to themselves and ask this question relative to the largest problem they face… or they’d like to get solved.
  5. Get into the Zone – True problem solving takes focused thought. Interruptions, phone calls, (cat coughing up a hairball – sigh), people dropping by, background noise… all reduce our ability to think deeply about a problem. When problem solving, stop the interruptions to the best of your ability (relax the cat is still breathing). Switch off the phone, hang up the do not disturb sign, go missing from action, (throw the cat (gently) into the backyard).

Some Mechanicals:
Intended to force structure on the Problem solving process.

  1. Don’t resist forces, use them – Put simply, the thing most preventing you from solving a problem is likely the best thing to use to solve the problem. This is more than the cliché of turning lemons in lemonade; it’s a way of looking at problems that more often than not leads to elegant solutions. Examples are all around us.
    • Bridges use gravity to hold everything together despite gravity’s ‘desire’ to pull everything down.
    • Politics regularly uses people’s self interests to get them to do what they wouldn’t normally do.
    • ‘Heat sinks’ in computers use the heat of the computer to create airflows which cool the machine.
    • ‘Pressure valves’ use excess steam pressure to vent excess steam pressure.

    Your starting point? List the forces relevant to the problem. You can’t start manipulating these until you know what they are. Making a list is the first step.

  2. Lists and Frameworks: #1 PESTLE
    PESTLE is nothing more than a mnemonic to remember possible ‘aspects’ affecting your problem/solution:

    When looking at a problem, how do the above areas of influence affect both the problem and your proposed solution? Perhaps more importantly? How will your solution affect these areas?

  3. Lists and frameworks: #2 SWOT analysis
    Take each point in the PESTLE analysis and go the next step; using the Strengths, Weaknesses, Opportunities and Threats. What additional insight can you gain about the situation?

    Item from PESTLE Strengths Weaknesses
  4. Running the Alphabet: (take 1)
    This might seems silly (so what?) but it’s a fairly effectively way of forcing you to think differently about your problem. Just make a list, A through Z and find something starting with that letter that relates to the problem either as a threat or opportunity. Use this when you’re stumped for ideas. It can’t hurt.
  5. Running the Alphabet: (take 2) (solution to a tiny, but common problem)
    Ever forget a word? You have it on the tip of your tongue but can’t ‘find’ it? Run through the alphabet slowly in your mind, pay attention to the ‘echoes’, does the letter ‘remind’ you of the word? You’ll find that you’ll know for certain that the word doesn’t begin with ‘S’ or ‘P’, others will raise a question in your mind. (as you get older, you’ll be using this more often… trust me)
  6. Time Management:
    Here’s a fact… you have a finite amount of time allocated to you.
    Here’s another fact… there are an infinite number of problems to solve.
    Hence an observation… you can’t solve them all … so how will you decide which ones to solve and which to ignore?

    You’re going to start with the ever present ‘to do’ list. (it exists even if you’ve been too lazy to write it down.)
    At the top of the list there are the things you must do!
    Next? The things you’ll do if you get the first section done.
    Then, there’s a list of things you won’t do/shouldn’t do/must stop doing.

    Remember Pareto’s Law: 80 % of your result is generated by 20% of your activities.

    Learn one simple word. NO! Use it when people expect you to do more than you have resources accomplish.

    Remember? You can’t do everything, so you have to decide what to do.

    Yes… this is old news. Yes it’s mundane. Yes it requires discipline. So?

  7. One last thought.
    Change one thing at a time. If you don’t, you won’t know which change caused the final result.


This LONG smorgasbord of ideas is not offered as a complete overview of problem solving. It was intended as a sampling of ideas, reasonably related to each other, presented to an audience of people with vastly different problem solving abilities and experience. My goal? Somewhere in the presentation is an idea new to each listener, something they can take back and use to good effect on their next, or their current problem.

© 2007 – 2015, Peter de Jager. All rights reserved.

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