*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).