Method 20

Vary the problem [7].

Use When

When there are very few signs of progress during the problem-solving process.  When your plan has proven difficult to follow, and alternative possible plans do not seem promising.

How

Sub-Method I)

1.  Start with modest variations of the problem first because it is more similar‡ to the original problem making it more likely for the similar problem to provide insight into the original problem.

Details of good strategies on how to vary the problem are listed below—most of which are suggestions given in [44] and [7] from George Polya, and Terrence Tao.

a)  Derive a consequence of the problem and solve that first [44].  In order to do this:

i)  Search for implications of the solution by using defeasible reasoning or†† experimentation (method 7).

ii)  Implement a fast and un-thorough problem-solving process on the new problem of solving for the new consequence found in (i).

iii)  If you receive sufficient insight to your problem, then choose your next heuristic for solving your original problem, otherwise go to sub-step (i) to search for another consequence of the original problem.

iv)  If you have done enough (hopefully not too much) work in (a), then go to step (b).

b)  Formulate a conjecture that implies the problem.  Solve that first [44].

        -You do this by experimenting (method 7) on concepts and information related to the problem.          Experiment on assumptions on this information to decide what implies your problem.

c)  Reformulate the problem [44].  For example, using such methods as contra-positive for certain proofs (method 36), or constructing the problem under a framework of another branch of mathematics (methods 10, 11). Consider what other more promising problem is equivalent to solving your problem.

d)  Consider special cases of the problem, especially extreme or degenerate ones [44].  This helps you find a general solution by revealing what specifications of the problem might be making the general problem difficult to solve.  This is due to the fact that solving the general problem has to solve all cases including those extreme or difficult cases.  (Go to method 22 to do consider special cases).

Other variations of the problem that Terrence Tao suggests trying are: generalize the problem (Go to method 23), solve a simplified version of the problem (Go to method 22), and examine solutions of similar‡ problems (Go to method 26) [44].

2.  Modify the problem significantly if step (1) doesn’t work [44].  Do this by using one or more of the following:

i)  Remove data [44].

ii)  Swap the data with the objective as your new problem [44].

iii)  Disprove your problem [44].

Sub-Method II)

Fields medal winner Terrence Tao says that one should:

1.  Vary the problem gradually in more extreme ways until the problem becomes obviously true (or easily solved) or obviously false (or obviously impossible to solve).  For example, you can do this by using sub-method (I).

2.  Try to figure out exactly what you did to the problem that caused the problem to break†, which will give you a clue on how to solve the main problem. (For more details on how to do this, see method 21)

3.  Analyze each extreme variation that makes the problem break†.

An example of doing this as explained by Dr. Tao:

“For instance, if you delete a hypothesis and then the problem becomes obviously false, then this clearly shows that the hypothesis must be used in an essential way to solve the original problem.  If you weaken a conclusion and the problem becomes obviously true, then you see that the main difficulty lies in whatever separates the stronger conclusion from the weaker one.  And so forth.”

Notes

†A problem "breaks" when you vary the problem and the problem becomes easily solved or impossible to solve, or obviously true or false (in the case of proving a theorem).  To weaken a conclusion means to make the conclusion less general.  (The word “break” as used by Terrence Tao.)

††  The word “or” in math is equivalent to the and/or that most people use in writing.  Here the use of “or” is inclusive rather than exclusive meaning that you can also both experiment and use defeasible reasoning.

‡Two entities, relationships, or etc. being "similar" means that many defining parts of the entities, relationships, etc. are equivalent or that the defining parts of the entities have equivalencies in aspects of the definitions that apply to defining the entities, relationships, or etc.

Method 21

Find the hard part of your problem [44]. This guides you where to experiment with entities in your problem and where to focus your attention during problem-solving.

Class of Method

A Method to Guide Your Control of Problem-Solving.  A Method for Diagnosis.

Use When

When your plan has proven difficult to follow, and alternative possible plans do not seem promising.

How

Sub-Method I)  (*)

1.  Vary your problem accordingly:  Weaken your conclusion or the requirements for your desired solution.  Gradually make the conclusion or solution requirements easier to solve for, until the problem †breaks.

2.  The part of the conclusion or desired solution that you varied when the problem †broke is the hard part of the problem to solve.

Sub-Method II)  (**)

1.  Vary your problem accordingly:  Gradually remove details of your hypothesis, or given information until your problem is obviously false.

2.  The combinations of parts of hypothesis or given information that you could remove that make the problem unsolvable are then essential to solving the problem.

3.  Take your original problem, and do (*).

4.  Use the problem-solving process on the broken problem in step (3), but let this problem now have a slightly stronger conclusion (or more general desired solution) than the broken problem from step (3). Incorporating this stronger part of the conclusion is where a difficulty lies in your problem.  Therefore you should focus your problem-solving on how to solve this difficult part of the problem.

5.  After solving this stronger problem, continue to gradually strengthen the conclusion to prove (or strengthen the generality of the desired solution) and repeat step (4).

6.  Taking into account your success or failure of applying this method, choose the next heuristic (method) you will use based upon good defeasible reasoning or by using the general method given on page 100.

Notes

†A problem "breaks" when you vary the problem and the problem becomes easily solved or impossible to solve, or obviously true or false (in the case of proving a theorem).  To weaken a conclusion means to make the conclusion less general.  (This word as used by Terrence Tao.)

Method 22

Solve special cases of your problem [7].

Class of Method

A Primary Problem-Solving Strategy.

Use When

When the problem proves difficult or complicated to solve.  This is a more time consuming and less promising method to use.  Also do this when more promising less time consuming heuristics have been tried, or especially when the overall problem is hard to solve, but most special cases are much easier to solve.  (Consider possibly using sub-method 22b before using this main method).

How

1.  Examine special cases of your problem that are easier to solve in order to hypothesize the solution that examples show to readily occur for your general problem.  Do this by following these steps:

        a)  Consider each entity type involved in your problem.  Specify such entities of your problem differently to create various special cases where in each special case, one entity in the problem is specified over potential classes (or types) of that entity from the general problem.  You can also do this to create other special cases where combinations of entities are fixed to specific entity sub-classes.

b)  Consider which of these cases seems easy to solve.  Usually it is better to consider first the cases you have resources to solve.

c)  Start problem-solving on each new promising special case.  When solving these special cases, try to solve them in a manner where steps and concepts used can be likely generalized for solving the original problem.  (See comment †).

d)  Until you have a sufficient variety of useful solutions of different special cases, return to and continue from sub-step (a) to find another useful special case of your problem if doing this isn't too difficult to do.

2.  ***Create the simplest explanation for the proposed solution to the problem that would hold on your example problems.  Do this by following these steps:

a)  Recognize all of the differences and similarities between solutions of the different special cases.

b)  Consider how each part of the differences between the special cases affected their differences of solution.        

c)  Consider which of the simplest general explanations for the effects between special cases would describe all of the special cases.

d)  Carefully but quickly consider if your explanation could be violated by other extreme or degenerate special cases.

e)  If you find an example that violates your explanation, then go back to sub-step (a) incorporating your new special case.

f)  Otherwise propose that explanation as your hypothesized solution to the problem.

3.  Do a quick search of counter-examples (method 8) for your hypothesized solution.

4.  If a counter-example is found, then continue looking for cases that would provide counter-examples, because if there is one, there are likely others.

5.  Go to step (2) and continue from there by incorporating the counter-examples as another special case.

6.  Otherwise, try to prove the solution hypothesized in step (2) by starting the problem-solving process on finding such a proof.

7.  If the proposed solution becomes difficult to prove, then do a more intensive search for counter-examples (Go to the counterexample sub-routine, method 8), or continue to experiment with special cases to try and generalize the solution further.

8.  Taking into account your success or failure of applying this method, choose the next heuristic (method) you will use based upon good defeasible reasoning or by using the general method given on page 100.

Notes

-If you do not find the solution to the problem through this method, then you will at least have acquired Information as to what does not work in your problem.  You will also have found what conditions must be met in your problem for many specifications of your problem, thus helping you restrict your search on how to solve your problem by abandoning methods or ideas that would contradict what you found in your examples.  In other words, applying this method to your problem has given you a feel for the problem that assists with future hypothesizing of a solution, and helps you to avoid wild goose chases.

†Perhaps in most cases where we unsuccessfully seek the answer to a question, the cause of the failure lies in the fact that problems simpler and easier than the one in hand have been either incompletely solved, or not solved at all.  Everything depends, then, on finding those easier problems and on solving them by means of devices as perfect as possible and of concepts capable of generalization.  –David Hilbert

Example

See example problem 13 of chapter 6 (Examples of Problem Solving).

Other Examples

-Examining limiting cases to explore the range of possibilities.

-Setting an integer parameter to 1, 2, 3... and looking for an inductive pattern (Go to method 12 to search for a pattern).

Sub-Method 22a

A Fast Version of: Solve special cases of your problem [7].  

Class of Method

A primary problem-solving strategy.

Use When

When you feel you don't understand the variety of details needed to solve the problem.  This is a routine used early in the problem-solving process to get a "feel" for the problem.

How

1.  Identify all aspects in which you could specify the problem to a special case.  Consider all available methods you already know on how to solve some of these special cases.

2.  Choose special cases of the problem where sufficient resources and common strategies most likely exist to solve the special case, and that would give you information you could likely hypothesize on or see a pattern on how to create a solution to the general problem.

3.  If your way of choosing special cases provides little to hypothesize on or recognize about the result applied to the original problem, then go back and choose another special case type that likely require different methods to solve.  Doing this is useful because different methods of solving examples could provide better insight for hypothesizing a general solution.

4.  Taking into account your success or failure of applying this method, choose the next heuristic (method) you will use based upon good defeasible reasoning or by using the general method given on page 100.

Notes

Using the primary method of solving special cases of your problem (method 22) may be too time consuming in many situations, and therefore proper control of the problem-solving process would suggest using this routine (method 22a) more often.

Sub-Method 22b

Consider what special case of your problem or other similar variant of your problem that if solved could be used as a main part of a plan for solving your original general problem, or for solving an important sub-problem. 

Class of Method

A Secondary Problem-Solving Method.

Use When

When applying more basic strategies and plans have not worked.  You might consider using this before the main method 22 to first solve those special cases that are more likely to help solve the problem.

How

1.  Specify various aspects of special cases or variations of your problem that you would likely have resources to solve.

  -You don’t want to specify special cases that are bizarre, have seemingly no usefulness, or that you wouldn’t likely have sufficient resources to solve.  This is because you need to have the special case assist you with your general problem.

2.  For each special case or variation, quickly try to solve your original problem or at least create a somewhat promising plan by using the assumed special case or variation as solved.

3.  Solve the special case or variation that seems most promising in both likelihood to be useful to solve your original problem, and the likelihood of being able to solve the special case.  If nothing seems very promising, then choose another heuristic (method).

4.  Restart the problem-solving process on solving your special case sub-problem.

5.  If your special case or variation proves difficult to solve, then return to step (3), otherwise, begin to solve the original problem more intensively by using the solution to the special case.

6.  Taking into account your success or failure of applying this method, choose the next heuristic (method) you will use based upon good defeasible reasoning or by using the general method given on page 100.

Example

See example problem 2 of chapter 6 (Examples of Problem Solving).

Another example is proving theorems for simple functions, and using those results to prove such theorems about more general functions.  Note that simple functions are functions that have finitely many values.

Method 23

Solve a more generalized version of your problem.  

Class of Method

A Problem Variation Sub-Method.

Use When

When specified parts in the given of your problem clutter or complicate the problem-solving process, and solving the problem by generalizing an entity of the problem simplifies these complications.

How

1.  Identify the entities in the problem that you can generalize that change the problem from one type of difficulty to another type of difficulty or lack thereof (You want another difficulty that you may better know how to deal with).

2.  Create your new problem as the generalization of your original problem by making the changes that allow you to work with difficulties that are more promising to overcome.

3.  Taking into account your success or failure of applying this method, choose the next heuristic (method) you will use based upon good defeasible reasoning or by using the general method given on page 100.

Notes

Here "generalized problem" means that you solve a problem that includes a wider class of problems that includes your specific problem.  For example, if your problem has a specified given value "x=12", you could generalize your problem by letting "x" be a variable instead of a specified value.