This book makes you think. Not that other books don't, but the authors present you with a continuous set
of problems and constantly ask you questions about how you are progressing. By the time you reach the end you are creating
elaborate models and making estimates that you would not dream of doing when you started.
If you are trying to solve a tough problem, chances are you are struggling with the creation of
a model to help you make progress towards solving it:
p.ix"Have you ever built a model? ...You would not be able to think
if you were incapable of building models. What you might not have done yet is build a model explicitly so that other
people can understand it and perhaps use it... An explicit model is a laboratory for the imagination. You can tweak a model
to see how it responds... You can explore its strengths and limitations. You can even guardedly make predictions and then
argue how good (or poor) those predictions might be."
Learning how to create models can help you to become a better problem solver.
p.x"We have written this book to demonstrate that there indeed is a subject called 'modeling' which
can be learned and needs to be learned. We believe that it is an important subject, not only because models are becoming so
pervasive, but also because the skills of modeling are so closely related to the more general skills of problem solving. Learning
to model is bound up with learning to solve problems and to think imaginatively and purposefully."
p.1"modeling is an integral part of problem solving in any discipline."
When you create a model, you separate the essential parts of the problem from the non-essential
parts and usually this helps you think about the problem in an isolated manner:
p.19"Whenever you build a model you have to be selective. You have
to identify those aspects of the real world that are relevant and ignore the rest. You have to create a stripped-down model
world which enables you to focus, single-mindedly, on the problems you are trying to solve."
Creating the model would serve no purpose unless we just sit and think about the model (the parts
and the interaction of the parts) for an extended period of time:
p.21"The point we want to make is that thinking consciously and explicitly about
models is a crucial part of problem solving in general."
Getting stuck is part of the problem solving process. It just means that we need to ask ourselves
a different set of questions:
p.28"Struggling is a precursor of learning and being temporarily perplexed is a natural phase in
problem solving."
If we can't come up with an exact solution, the chances are good that we can easily create upper
bound and lower bound estimates:
p.45-46"As we have seen, searching for simplified versions of the
problem, or for upper and lower bounds, can also lead to the solution, albeit in two stages. The first stage is to find a
simple model or a bound. The second stage is to ask how we can refine that model or bound... Powerful heuristics
generate useful ways of thinking... We solve problems by thinking. It seems as though people need stimuli to provoke thought
or to make their cognitive leaps. [problem solving] Heuristics are powerful stimuli... Asking how we recognize that a heuristic
is pointing in a useful direction or how we know that we are using heuristics intelligently are both much more difficult questions.
There are no easy answers and there are certainly no short answers. If we could answer these questions in a paragraph or two,
there would be no need for this book."
We can even think about the process of cooling a beer in a refrigerator and create models of the
process. A search of the Internet will reveal that the subject of cooling a warm beer has been the subject of much research
(!). Would you believe that due to gravity-based convection currents, a beer cools faster by lying on its side (rather than
standing straight up on the shelf). There is even a device you can buy that spins a beer can in a small tank of ice-water,
providing rapid cooling as the warm beer within the can is rapidly brought into contact with the cooler aluminum can.
p.60"[equation for modeling the temperature of a warm beer cooling in a refrigerator, where T is
the temperature of the beer and F is the temperature of the refrigerator]
T n+1 = Tn - K dt (Tn - F)"
The authors tell us to keep on track by asking ourselves questions that we should be able to answer.
This way we can check to see if we are headed in the right direction:
p.152"If you think you know what you are doing, ask yourself a question
or set yourself a task to convince yourself that you know what you are doing!"
Once again, the authors provide a large number of questions that serve as a frame work for thinking.
The problems we face must be solved by thinking, so having a framework for our thoughts is often a good idea.
p.160-164"What information do we have? ...What assumptions can we make? ...What, in particular,
would we like to know? ...We do not know the answers to these questions, but can we make any reasonable assumptions? ...Is
there a simpler version of the problem that we could think about? ...Have we thought how the element of chance affects our
problem? ...How can we express the likely outcome of following one strategy rather than the other? ...What kind of model
could we build? How would we use it?"
