*This page written by Dan Styer, Oberlin College Physics
Department;http://www.oberlin.edu/physics/dstyer/SolvingProblems.html;last
updated 9 January 2002.*

To set the stage, I want to discuss an example of problem solving from
everyday life, namely building a jigsaw puzzle. There are a number of different
approaches to building a jigsaw puzzle: My approach is to first turn all the
pieces face up, then put together the edge pieces to make a frame, then sort the
remaining pieces into piles corresponding to small "sub-puzzles" (blue pieces
over here, red pieces over there). I build the sub-puzzles, then piece the
sub-puzzles together to build the whole thing. Other people have different
approaches to building jigsaw puzzles, but nobody, *nobody*, builds a
puzzle by picking up the first piece and putting it in exactly the correct
position, then picking up the second piece and putting it in exactly the correct
position, and so forth. Solving a jigsaw puzzle involves an approach--a
strategy--and a lot of "creative fumbling" as well.

Your physics textbook contains many solved "sample problems". The solutions presented there are analogous to the completed jigsaw puzzle, with every piece in its proper position. No one solves a physics problem by simply writing down the correct equations and the correct reasoning with the correct connections the first time through, just as no one builds a jigsaw puzzle by putting every piece in its correct position the first time through. The "solved problems" in your book are extraordinarily valuable and they deserve your careful study, but they represent the end product of a problem solving session and they rarely show the process involved in reaching that end product. This document aims to expose you to the process.

Solving a physics problem usually breaks down into three stages:

- Design a strategy.
- Execute that strategy.
- Check the resulting answer.

If you are looking for a child lost in the woods, your first step is to sit down, think about what the child probably did and where he probably is, and devise a strategy that will allow you to effectively rescue him. If, instead, you just rush about the woods in random directions, you're likely to become lost yourself.

**Where are you now, and where do you want to go?** Before you can design
a path that takes you from the statement of the problem to its answer, you must
be clear about what the situation is and what the goals are. It often helps to
*check off* each given datum of the problem, and to *underline* the
objective. But for getting an overall sense of the problem, nothing beats
summarizing the whole situation with a diagram. The diagram will organize your
work and suggest ways to proceed. One of my course graders told me that "When
students draw a diagram and label it carefully, they are forced to think about
what's going on, and they usually do well. If they just try a globule of math,
they mess up."

**Keep the goal in sight.** Don't get caught in blind alleys that lead
nowhere, or even in broad boulevards that lead somewhere but not to where you
want to go. It sometimes helps to map a strategy backwards, by saying: "I want
to find the answer *Z*. If I knew *Y* I could find *Z*. If I knew
*X* I could find *Y* . . . " and so forth until you get back to
something you are given in the problem statement.

Some students find it useful to make a list of the information given and the goal to be uncovered (e.g. "given the constant acceleration, the initial velocity, and the time, find the displacement"). Others find it sufficient to write down only the goal (e.g. "to find: displacement").

**Ineffective strategy.** Do not page through your book looking for a
magic formula that will give you the answer. Physics teachers do not assign
problems in order to torture innocent young minds . . . they assign problems in
order to force you into active, intimate involvement with the concepts and tools
of physics. Rarely is such involvement provided by plugging numbers into a
single equation, hence rarely will you be assigned a problem that yields to this
attack. In those rare instances when you do face a problem that can be solved by
plugging numbers into a formula, the most effective way to find that formula is
by thinking about the physical principles involved, not by flipping through the
pages in your book.

**Make the problem more specific.** You're asked to find the number of
ways that *M* balls can be placed into *N* buckets. Suppose you can't
even begin to map out a strategy. Then try the problem of 3 balls in 5 buckets.
Solving the more specific problem will give you clues on how to solve the more
general problem. And once you use those clues to solve the more general problem,
you can check your solution by trying it out for the already-solved special case
*M*=3 and *N*=5.

**Large problems.** At times you will be faced with big problems for which
no method of solution is immediately apparent. In this case, break your problem
into several smaller subproblems, each of which is simple enough that you know
how to solve it. At this strategy-design stage it is not important that you
actually solve the subproblems, but rather that you know you can solve them. You
might begin by mapping out a strategy that leads nowhere, but then you haven't
wasted time by implementing this strategy. Once you have mapped out a strategy
that leads from the given information to the answer, you can then go back and
execute the calculations. This strategy has been known from the time of the
ancients under the name of "divide and conquer".

**Work with symbols.** Depending on the problem statement, the final
answer might be a formula or a number. In either case, however, it's usually
easier to work the problem with symbols and plug in numbers, if requested, only
at the very end. There are three reasons for this: First, it's easier to perform
algebraic manipulations on a symbol like "*m*" than on a value like "2.59
kg". Second, it often happens that intermediate quantities cancel out in the
final result. Most important, expressing the result as an equation enables you
to examine and understand it (see the section on "Answer Checking") in a way
that a number alone does not permit.

(Working with symbols instead of numbers can lead to confusion as to which symbols represent given information and which represent unknown desired answers. You can resolve this difficulty by remembering--as recommended above--to "keep the goal in sight".)

**Define symbols with mnemonic names.** If a problem involves a helium
atom colliding with a gold atom, then define *m _{h}* as the mass of
the helium atom and

**Keep packets of related variables together.** In acceleration problems,
the quantity (1/2)*at*^{2} comes up over and over again. This
collection of variables has a simple physical interpretation, transparent
dimensions, and a convenient memorable form. In short, it is easy to work with
as a packet. Take advantage of this ease. Don't artificially divide this packet
into pieces, or write it in an unfamiliar form like
*t*^{2}*a*/2. Packets like this come up in all aspects of
physics--some are even given names (e.g. "the Bohr radius" in atomic physics).
Look for these packets, think about what they are telling you, and respect their
integrity.

**Neatness and organization.** I am not your mother, and I will not tell
you how to organize either your dorm room or your problem solutions. But I can
tell you that it is easier to work from neat, well-organized pages than from
scribbles. I can also warn you about certain handwriting pitfalls: Distinguish
carefully between *t* and *+*, between *l* and 1, and between
*Z* and 2. (I write a *t* with a hook at the bottom, an *l* in
script lettering, and a *Z* with a cross bar. You can form your own
conventions.) These suggestions on neatness, organization, and handwriting do
not arise from prudishness--they are practical suggestions that help avoid
algebraic errors, and they are for your benefit, not mine. (On the other hand,
it doesn't hurt to be neat and organized for the benefit of your grader. One
course grader of mine pointed out: "If I can't read it, I can't give you
credit.")

**Avoid needless conversions.** If the problem gives you one length in
meters and another in inches, then it's probably best to convert all lengths to
meters. But if all the lengths are in inches, then there's no need to convert
everything to meters--your answer should be in inches. In fact, you might not
actually need to convert. For example, perhaps two lengths are given in inches
and the final answer turns out to depend only on the ratio of those two lengths.
In that case, the ratio is the same whether the lengths going into the ratio are
inches or meters. It's easy to make arithmetic errors while doing conversions.
If you don't convert, then you don't make those errors!

**Keep it simple.** I will not assign baroque problems that require
tortuous explanations and pages of algebra. If you find yourself working in such
a way, then you're on the wrong path. The cure is to stop, go back to the
beginning, and start over with a new strategy. (Generations of students have
kept track of this rule by remembering to KISS: Keep It Simple and
Straightforward.)

**Dimensional analysis.** Suppose you find a formula for distance (in,
say, meters) in terms of some information about velocity (meters/second),
acceleration (meters/second^{2}), and time (seconds). If your formula is
correct then all of the dimensions on the right hand side must cancel so as to
end up with "meters".

**Numerical reasonableness.** If your problem asks you to find the mass of
a squirrel, do you find a mass of 1,970 kilograms? Even worse, do you find a
mass of -1,970 kilograms?

[**Reasonable speeds.** "My calculations give me a speed of 23 m/s. Is
this reasonable?" It's hard for most people to get a feel for the reasonableness
of speeds expressed in meters per second. Until this qualitative feel develops,
Americans should check for reasonableness by converting speeds in meters per
second to speeds in miles per hour: simply double the number (20 m/s is about 40
mi/hr). Non-Americans should convert to kilometers per hour: simply quadruple
the number (20 m/s is about 80 km/hr).]

**Algebraically possible.** Would evaluating your formula ever lead you to
divide by zero or take the square root of negative number?

**Functionally reasonable.** Does your answer depend on the given
quantities in a reasonable way? For example, you might be asked how far a
projectile travels after it is launched at a given speed with a given angle.
Common sense says that if the initial speed is increased (keeping the angle
constant) then the distance traveled will increase. Does your formula agree with
common sense?

**Limiting values and special cases.** In the projectile travel distance
problem mentioned above, the range is obviously zero for a vertical launch. Does
your formula give this result? If you solve a problem regarding two objects,
does it give the proper result when the two objects have equal masses? When one
of them has zero mass (i.e. does not exist)?

**Symmetry.** Problems often have geometrical symmetry from which you can
determine the direction of a vector but not its magnitude. More often they have
a "permutation" symmetry: If your problem has two objects, you can call the cube
"object number 1" and the sphere "object number 2" but your final answer had
better not depend upon how you numbered your objects. (That is, it should give
the same answer if every "1" is changed to a "2" and vice versa.)

**Specify units.** "The distance is 5.72" is not an answer. Is that 5.72
miles, 5.72 meters, or 5.72 inches? Similarly, if the answer is a vector, both
magnitude and direction must be specified. (The direction may be drawn into a
diagram rather than stated explicitly.)

**Significant figures.** Any number that comes from an experiment comes
with some uncertainty. Most of the numbers in this course come with three
significant figures. If a ball rolls 3.24 meters in 2.41 seconds, then report
its speed as 1.34 m/s, not 1.34439834 m/s. Most introductory physics courses do
not require a formal or technical error analysis, but you should avoid
inaccurate statements like the second quotient above.

**Large problems.** If you break up your large problem into several
subproblems, as recommended above, then check your results at the end of each
subproblem. If your answer to the second subproblem passes its checks, but your
answer to the third subproblem fails its checks, then your execution error
almost certainly falls within the third subproblem. Knowing its general
location, you can quickly go back and correct the error, so its effects will not
propagate on to the remaining subproblems. This can be a real time-saver.

- Strategy design
- Classify the problem by its method of solution.
- Summarize the situation with a diagram.
- Keep the goal in sight (perhaps by writing it down).

- Execution tactics
- Work with symbols.
- Keep packets of related variables together.
- Be neat and organized.
- Keep it simple.

- Answer checking
- Dimensionally consistent?
- Numerically reasonable (including sign)?
- Algebraically possible? (Example: no imaginary or infinite answers.)
- Functionally reasonable? (Example: greater range with greater initial speed.)
- Check special cases and symmetry.
- Report numbers with units specified and with reasonable significant figures.

- George Polya,
*How To Solve It*(Princeton University Press, Princeton, New Jersey, 1957).

- Donald Scarl,
*How To Solve Problems: For Success in Freshman Physics, Engineering, and Beyond*, third edition (Dosoris Press, Glen Cove, New York, 1993).

- James L. Adams,
*Conceptual Blockbusting: A Guide to Better Ideas*(Norton, New York, 1980), - Berton Roueche,
*The Medical Detectives*(Times Books, New York, 1980) and*The Medical Detectives, volume II*(Dutton, New York, 1984), - Martin Gardner,
*Aha! Insight*(Freeman, New York, 1978), - Donald J. Sobol,
*Two-Minute Mysteries*, - Arthur Conan Doyle, Sherlock Holmes stories,
- Agatha Christie, Hercule Poirot stories, particularly
*Murder on the Orient Express*.

- Frederick Reif, "Understanding and teaching important scientific thought
processes",
*American Journal of Physics***63**(1995) 17-35 (especially section V), - Rolf Plotzner,
*The Integrative Use of Qualitative and Quantitative Knowledge in Physics Problem Solving*(Peter Lang, Frankfurt am Main, 1994).