Consider the following parable:
Once upon a time there were two farmers, Simplicio and Salvatio. They were very similarly situated. They operated neighboring tomato farms. Their fields were nearly the same size.
On harvest day, Simplicio said “I’m going to work really hard today. I’m going to work so hard that I harvest twice as many tomatoes as the other guy. I planted twice as many plants, so there should be no problem. I’m going to turn a huge profit.”
However, there is one slight problem. Because the plants were too close together, they weakened each other. Then the worms came along and finished them off. As a result, there are hardly any tomatoes in Simplicio’s field. It doesn’t matter how hard he works on harvest day; he can’t harvest tomatoes that aren’t there.
Problem-solving day is like harvest day. You get to harvest and profit from facts and ideas that are in your brain. The process is subject to a crucial constraint: You cannot harvest stuff that isn’t there. If you didn’t put ideas into your head in the proper way, you won’t be able to get them out when needed, no matter how hard you work at it.
Let’s be clear: In large measure, your success at problem-solving depends on what you did during previous days, months, and years.
When reading this document – or any other writeup on problem-solving – please be aware of its limitations. There is nothing wrong with what is said here; the problem is what is not said. There is some value in problem-solving techniques per se, but you must look elsewhere to find the other 95% of what you need for effective learning, thinking, and problem-solving.
Here’s another limitation: A checklist such as this one can be useful as a reminder of things you already mostly know. However, it’s hard to learn new ideas from a checklist. It’s too condensed. In some cases, an individual item on this list (e.g. item 46) would become a book chapter or an entire book if spelled out. If you want to learn, it helps to have more detail. It helps to see the ideas applied to examples. There are quite a few good examples in reference 1. An additional nice example is discussed in reference 2.
This applies to every new idea you encounter, whether it comes from something you read, or an experiment you did, or a calculation you did, or whatever. This is discussed in some detail in reference 3.
The connections are necessary if the idea is to be useful, because they are what allow you to recall the idea when needed. To say the same thing the other way, if you put an idea into your brain but have no way of recalling it when needed, it might as well not be there. An idea that can be recalled in many ways is vastly more useful than one that can only be recalled in one way.
This is important for two reasons:
In more detail:
Following this advice may slightly increase the workload in the short run, but it greatly reduces the workload in the long run, especially when dealing with complex problems. For details on this, see reference 4.
Checking the work at each step along the way greatly increases the odds of getting the right answer. In a complex calculation, there will be mistakes, and showing the work makes it easier to find and fix the mistakes. Showing the work and showing the checks makes it easier to check and re-check the work. It is particularly valuable in a teamwork situation, because it helps other people check the work and learn from the work. For more on this, see section 4 and item 15.
This is particularly important when you are working as part of a problem-solving team. You are free to make up whatever notation and terminology you like, but don’t expect other people to guess what it means.
There exist thousands of different types of diagrams; reference 6 lists more than 100 examples. If you cannot find a pre-existing way of representing the data, invent something. If this is not easy, do it anyway. Remember the sardonic proverb:
Procedures for dealing with ill-posed problems are discussed in reference 7.
You are absolutely not required to fully understand every element of what’s going on before starting the analysis and calculations. It is common to understand a problem well enough to solve it, and then achieve a much deeper understanding by pondering the solution. See also the discussion of formalism in section 5.
Analysis helps with the understanding ... just as understanding helps with the analysis. You can leverage one against the other, again and again, recursively and iteratively, like the itsy-bitsy spider.
Even though it is sometimes possible (as mentioned in item 11 and item 12) to obtain a formally-correct result without understanding it, this should be considered only a first step. It is not something to be proud of. Don’t walk away from the result until you have made a serious effort to understand it. See also item 14 and item 16.
Example: There are two remarkably dissimilar solutions to the Mississippi Flow problem (reference 3).
Example: There are two remarkably dissimilar solutions for finding the Johnson noise in a parallel combination of resistors (reference 8).
Obviously this overlaps quite a bit with the suggestion to Check The Work (item 4) but it is worth mentioning, because it is a specific way of checking the work. The obvious approach is to carry out the same calculation a second time, checking and re-checking each step along the way.
The non-obvious approach is to carry out a completely different calculation that produces the same result. This has the tremendous advantage that it can catch conceptual errors, not just arithmetic errors. (The downside is that if there is an error, this will not tell you exactly where to find it.)
Simple example: Check by doing the inverse problem. Given the task 10 − 7 = ____, the answer plus 7 had better equal 10.
Intermediate example: If you use trig functions to find the sides of a triangle, the answer had better uphold the Pythagorean theorem and other trig identities. Similarly, any physics answer had better uphold the various conservation laws.
Fancy example: Given the task of estimating (closed book) how much Mississippi River water flows past New Orleans in a year, there are at least two completely different ways of approaching the problem, leading to completely independent estimates.
For students who have never seen the problem before, finding one solution is hard ... and finding two independent solutions is even harder, much worse than twice as hard.
Here’s an example that recently came across my desk: Suppose you have to multiply (20.14 + 20.04) × (20.14 − 20.04). You could grind it out, and obtain 4.018. On the other hand, if theory tells you that 20.14 is really 2cosh(ρ) and 20.04 is really 2sinh(ρ) where ρ=3, then you know the answer has to be 4, exactly, independent of ρ. So that’s a solution to the current problem and innumerable others, because it works for all ρ.Furthermore, you get to go looking for a good physics reason why the answer had to be independent of ρ, which makes you smarter in the long run.
What do we mean by analogous? That could be almost anything you know that has some features in common with the current problem. Here are some hints on places where you might look for analogies and/or approximations you can introduce:
There are lots of ways of doing this: Taylor series, Fourier series, et cetera. For example, the sphere mentioned in the previous item is the lowest-order term in an expansion based on spherical harmonics.
Sometimes your first approxmation turns out to be good enough. If not, maybe you can redo the analysis using slightly better approximations. Sometimes the approximation is off by a factor of infinity and cannot be repaired, but presumably you learned something from the process of doing the analysis. Remember the spirit here try something and see if it works. If not, try something else.
The difference between an approximation and an analogy is that you don’t even pretend that the analogy is the right answer. After doing the analogous problem, you set it aside and start attacking the original problem, using what you learned from the analogy as a roadmap.
For example, balancing a complicated chemical reaction equation would be well-nigh impossible if you didn’t have a systematic way of doing it. Gaussian elimination might require 20 steps or even 100 steps, but each step is cut-and-dried and incredibly easy, so you just grind it out. Or tell the computer to do it. Problem solved.
Feynman compared knowledge to a grand tapestry. Something you don’t know is like a hole in the tapestry. You can fix it by weaving upward from the bottom, downward from the top, and/or inward from the sides. Sometimes it pays to start in the middle of nowhere and construct something from scratch, and then patch it into the main tapestry later.
The straight-backward approach works particularly well for some of the multiple-guess tests that plague our classrooms today, but does not work nearly so well in the real world.
Rationale: In the best case, erasing isn’t the trouble. In the worst case, you run the risk of erasing something valuable.
The “scientific method” is not nearly as methodical as non-scientists seem to think it is.
Thinking of it as a step-by-step process suggests stepping along a road, moving steadily forward along a one-dimensional path. That’s not how problem-solving actually works. That may be a good way to explain the solution once it is found, but that’s not a good way to find it.
The reality is more like searching a maze (rather than following a clear one-dimensional path). It goes something like this:
Start with an intuitive idea. Draw a qualitative diagram. That provides motivation to write some equations, which can be turned into software code, which produces a vastly more accurate quantitative diagram. Looking at this diagram indicates that the original intuitive idea is not entirely satisfactory. Checking the limiting cases shows that the idea works in certain limits, but not in the general case. This suggests a more elaborate idea. That motivates writing some new equations and some new code. That produces a new diagram. Checking the limiting cases and the symmetries suggests that the “elaborate” model can be made much more streamlined and more intuitive. That motivates yet more equations. Alas the equations show that the streamlined model is not actually correct. So we have to back up and come up with yet another model. A new concept is brought to bear. It is sophisticated but not complicated. This motivates an entirely new approach, including new equations, new code, and new diagrams. The result is elegant and provably correct. It is intuitive in retrospect. There is an algebraic proof and a non-algebraic geometric proof, which can be checked against each other. An analysis of the symmetries, limiting cases, and analogies makes it even more plausible and intuitive.
One lesson to take from this story is that a check-list is not a do-list. You should check each of the items enumerated in section 2, but you don’t necessarily need to do them all. You certainly don’t need to do them in any particular order. Some of them will get done multiple times at various stages of the game.
Very commonly you chip away at one corner of the problem and go as far as you can until you get stuck. Then you set that aside and start chipping away at another corner.
Very commonly a seemingly-promising line of attack turns out to be a dead end, and a big chunk of preliminary work has to be abandoned. Abandoning something is painful and difficult for psychological reasons, but it has to be done.
Consider the following styles:
For most of the last 2300 years, there has been a tendency to pretend these are all the same thing ... when in fact they are not. Not even close.
I like elegance and polish as much as the next guy, but not when it requires pulling rabbits out of hats in such a way that the students cannot understand the process.
There will always be enormous temptation to polish things, for some good reasons and some not-so-good reasons.
The polished method works fine if you are using the solution for the third or fourth time. | The polished method isn’t at all representative of the process of discovering the solution for the first time. |
If all you want is a solution, the elegantly polished solution is fine. | If you want to learn how to solve problems, you need to see the problem-solving process, which is rarely (if ever) elegant. |
When teaching, or when writing a textbook, you have to decide whether you are training solution-users or educating solution-discoverers. It isn’t practical to show the gory details every time, but I suggest showing them at least occasionally.
There’s even a name for this: hindsight bias. That means you are much more likely to see how somebody thinks they should have solved the problem, not how they really solved the problem.
Let’s follow up on the idea of checking the work, as mentioned in item 4.
Solving a hard problem often includes a process that is equivalent to searching a maze. Mathematically this corresponds to enumerating the corners of a high-dimensional hypercube. In computer science, it is called walking the tree. In any case, when you return to a given node, it is important to know what’s been tried before; otherwise you risk gross inefficiency, up to and including infinte loops. Therefore you need to leave bread crumbs to mark the path.
This is super-extra important in conjunction with item 32.
An example of a relatively simple problem that rewards systematic tree-walking is Cheryl’s Birthday. A more elaborate example is Who Owns the Fish (reference 10).
Also: Inspiration is a way of obtaining an answer very quickly ... but beware that all too often, it is a way of obtaining a wrong answer very quickly.
For example, if you prove that a^{2} + b^{2} = c^{2} for «all» triangles, you’ve proved something that isn’t true. The statement should be restricted to right triangles in the Euclidean plane ... and your proofe of the statement should be clearly dependent on these restrictions.
Maybe for an easy, non-tricky problem you can get away with solving it only once, but if you are dealing exclusively with easy, non-tricky problems you wouldn’t be reading this document.
This is of course related to item item 4: check the work.
Example: South-West-North equilateral triangle on the surface of a sphere.
Formalism is a tool. Any tool can be abused ... but that doesn’t mean you are obliged to abuse it. Powerful tools, when properly used, make it easier (not harder) to understand what’s going on. For example: Consider the following contrast:
Maxwell’s paper on electrodynamics is almost unreadable. That’s because he didn’t use vector formalism. Vectors were not invented until many years later. | Nowadays, we write the Maxwell equations in terms of vectors ... or, preferably, bivectors. |
When you see the vector equation, you need to know the vector formalism to make sense of it. |
Without vectors, you are left with a bunch of dry mathematical equations, and figuring out what they mean physically is exceedingly difficult. | Using vector ideas, you can visualize what’s going on. The vectors have a physical significance. |
If you try to do without vectors, you wind up with so many equations it’s difficult to keep track of them. | The vector equations are quite compact. You can look at them all at once. |
So, when trying to solve an electrodynamics problem, there are three possibilities:
Here’s another example that supports the same conclusion:
If you learn to play the piano, in the sense of hitting hitting the right notes, it does not automatically bestow artistry or a “feel” for the music. | Learning to play the piano does not prevent you from acquiring artistry and musical feeling, and may even help. |
After all, you could program an unfeeling robot to hit the right notes. | Just because you “could” play like a robot doesn’t mean you are obliged to. |
In all cases, remember the proverb:
I say learn to use the tool, and then do something sensible with it.