solving problems

When I was a junior undergrad - sometime in the late Pleistocene -  I decided to write down what I had learned about problem solving. It has a bit of a Polya-ish flavor, but it reflected the bits and pieces of what I had learned.  I still keep a copy over my desk.  Some of the pieces, particularly those on creativity, mean much more to me now and I would go on for pages, but just seeing them calls all of that up. 

 

Problem selection is more of an art and I have never written much about it as new problem spaces tend to only present themselves when you are hard at work on something else that is at last partly related.  I have never had a problem with finding a list of great problems to work on …  a larger issue is which ones should I be working on.  That can have different answers depending on what the goal is (who or what I'm working for)

 

I was a physics and math double major, but I've found it useful in many other areas.  I should probably write a bit about problem selection other than picking something that you might have a bit of hope on gaining a bit of understanding.

Of course there are many other approaches - this is just mine. Not every problem would have all of these steps.  

 

So here goes: 

1. Understand the problem

 

- do you understand all of the words and equations in the problem?

- can you restate the problem in your own words?

- can you think of a picture or diagram to help understand the problem?

- what do you want to find or show?

- what level of accuracy do you need?

- is there enough information to find a solution?

- is the problem a relevant problem or is it gibberish?

- what is the scale of the problem?  can you do it in an hour or will it require unobtainium?

2. Devise a plan

 

- guess and check

- consider special cases.  what are the boundary conditions?

- select your tools

- do you fully understand your tools?

- draw a picture

- solve a simpler version first

- create a model

- use approximations - but understand their impact

- work backwards

- be ingenious …  this comes with experience and often with re-thinking the problem in a different context.  

This and the next step - connecting the dots are the non-trivial piece!!  Remember:

play and your ability to fail with grace are essential

curiosity is your driver - remember informed ignorance rules in science and math! never lose fact that your final job is to expand your ignorance - results are a by-product!!

- connect the dots - this is a subset of being ingenious

serendipity is another driver.  remember that serendipity is making discoveries by accident, but you must have enough sagacity to note what holds potential.  

dot connecting is best done using averted attention.  Don't tool constantly on your problem.  Focus hard, but pivot off to equally long period of something completely different. This may even fool others into thinking you have a life. 

3. Carry out your plan

 

- find a quiet place with no interruptions.  nature is your friend

- use some care and be a craftsman

- if you get stuck, take some time off and try to forget about what you are doing

- if you fail, use another approach

- know when you have failed

- if you are working with others know where the strengths are and divide appropriately

4. Check with reality

 

- does your answer make sense?

- can your solution make testable predications?

- ask an expert in your field to examine your work for flaws

- reflect on what you have done and examine was worked and what didn't

 

5. Next steps

- what are the new areas that expose new and even beautiful ignorance on your part?  These may be diamonds on the beach to explore.  Perhaps even a new beach and that is your reward.

 

_____________

A Recipe

Cherry and Chocolate Scones

Ingredients

For the scones:

 
° 500 g  all-purpose flour (4 c)

° 55 g white sugar (1/4 c)

° 8 g baking powder(1-3/4 tsp)

° 5 g baking soda (1 tsp)

° 4 g kosher salt (1-1/2 tsp)

° 250 g unsalted butter, cut into half inch pieces (2¼ sticks)

° 300 g cups heavy cream (1-1/4c)

° 125 g dried cherries (1-1/2c)

° 150 g chocolate chips (1 c)  depending on your taste regular Hershey's or the good stuff.  I use a mixture of chocolate chip sized quality milk chocolate and half 60% chocolate

 

For the glaze

 
° 85 g confectioners’ sugar (2/3 c)

° 2 tbl skim milk

° 1 tbl cherry puree

 

Technique

 

° Preheat oven to 350°F with racks in upper and lower thirds.

° Whisk flour, sugar, powder, soda, and salt in a large mixing bowl. Using a pastry cutter, cut in butter until combined with some pea-sized lumps. Stir in cream, cherries, and chocolate chips until combined.

° Put quarter cup sized scoops of dough on 2 buttered baking sheets and bake, switching sheets half way through (and you were worried about the rack arrangement:-), until golden and cooked through, something like 15 to 20 minutes.

° With the scones baking, make the glaze by whisking together sugar, milk, and cherry puree in a small bowl.

° Once scones are cooled, spoon over glaze, set and cool to about room temperature if you can stand the anticipation, and then scarf down.

 

Comments

[Jean Russell]

One of the treasures I hold from our delightful conversations is the clarity that serendipity is not only the surprising connection - it is also the awareness of the significance of it. thus developing skills around serendipity not only involves creating conditions for random unexpected connection - it is also developing wisdom about what is significant and doing a lot of noticing and reflecting. #gratitude

Love that you shared your approach to playing with problems.

[Charlie Hess]

Steve, thanks for those thoughts. The aspects I have recently come to appreciate are faith and openness. I had ( he died unfortunately for me) a friend who would talk about Setting the Conditions for Grace to Arise. I have found that making a conscious decision to be open and alert to all sorts of guidance is a wonderful way to set those conditions...and for problems to be approached. When I do mind maps with people , which help explore their lives and what is , I have such a sense of guidance and I really have faith in that even when I am led to places my mind would not have gone. I would say it ALWAYS is helpful in a big way.
So I would add faith and openness as two tools to your list.
Also, I would be very interested in how you see expanding ignorance in its various aspects...if you have time and inclination. thanks Steve!

[Jheri]

You gave this list to me a few years ago when I started back into school. Thankyou for it! It has been more useful than you can imagine:) And thank you for the wonderful exercises you gave me for my differential equations class:) Seeing them as art taught me so much.

[Jheri]

Ooooh those scones look wonderful too! I need to try them even if I'm not a good baker. Chocolate and cherry are wonderful together.