Method 1

Focus on what you want.

Class of Method

A Primary Problem-Solving Strategy.

Use When

You always focus on what you want because finding what you want is the ultimate goal of your problem.  Do not do anything that you do not think would likely assist with helping you find what you want.

How

1.  Understand what you want.  Give what you want a name by creating proper notation (method 5).

2.  Analyze what you want.

-Search for equivalents to what you want.

-Consider what causes could or seem† to produce such a result or implication, and where you have seen such a result produced.  Each of these is called a Circumstance Cause.

-Consider what others have done to obtain similar types of results.

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

† Whenever words like “seems” or “intuitively” are used, it means that you use your defeasible reasoning to make such judgments.

Method 2

Relate what you know to what you want.

Class of Method

A Primary Problem-Solving Strategy.

Use When

One of the first strategies to implement.

How

1.  Search for unknowns/objectives that are equivalent to finding/achieving your unknown/objective.  Each of these is called an Equivalent.  Sub-methods 2a and 2c can assist with finding equivalents.  Of course sometimes there are no obvious or useful equivalents, and you simply work with your unknown/objective.

2.  For your unknown/objective, and for each equivalent unknown/objective, consider what circumstances usually† cause such a result or a similar result.  Each of these is called a Circumstance Cause.  Method 1, and sub-methods 2b and 2c can assist in finding these.

You may even consider working backwards further (sub-method 2d) using defeasible reasoning to make larger steps in the backwards direction from an equivalent where you can solve the smaller details of this large step later.   Here you would work backwards from your circumstance cause to find even other potential circumstance causes.  You continue to work backwards as long as there are not too many possible branches to solve your problem in the backwards direction.  As soon as there are too many branches, you stop and remember your potential branches you have considered in the backwards direction.

3.  Consider what usually or definitely is caused by the given in the problem (and implied together from other resources).  Each of these is called a Likely Implication.  You may even need to consider “likely implications” of “likely implications” together with resources and the given.  Doing this (step 3) is called “working forwards”.

Be sure to use all of the data that seemed important or necessary for solving the problem when doing this step.  When searching for useful likely implications or resources, consider what theorems relate parts of the data in a form seemingly more related to what you want.  Search for theorems that would apply the data or given that produces information or implications that have likely useful similarities to the parts of what you want.

If it is hard to find likely implications, then consider the implication of each part of the given information in the problem in carefully selected combinations (potentially every combination) with other parts of the given information in the problem.  Do this using defeasible reasoning first, and then in more detail if seemingly useful to do so.  In order to do this, you may need to do search for patterns (method 12) to find more resources and likely implications from the given in the problem.

4.  Consider what from step 3 (likely implications) could† cause either an equivalent from step 1, or cause a "circumstance cause" found in step 2.  Each of these is called an Intermediate Step.  Sub-method 2c can assist in finding intermediate steps.  If you have a hard time finding a promising “intermediate step”, then you return to 2 and 3 to work forwards and backwards further in order to return to this step (4) with more potential connections to potential solution paths for an intermediate step.

5.  Your new relate plan is to solve the sub-problems of achieving each step of finding the most promising ways to link each step of the following process or any simplification of that process to achieve your unknown:  

Given information and resources → a likely implication → an intermediate step (if necessary) → a circumstance cause → an equivalent → your unknown.

Of course, sometimes your process can be more complicated, such as:

Given information and resources → a likely implication → (potentially likely implications from likely implications) → an intermediate step (if necessary) → another circumstance cause from working backwards → ....working backwards.... → a circumstance cause → an equivalent → your unknown.

Achieving each of the steps above is a sub-problem that you have to solve by potentially using the same type of “relate” strategy.

6.  If you failed to find a promising strategy to solve the problem, then you may find it necessary to restart from step 1 to search for more equivalents, circumstance causes, likely implications, and potential intermediate steps.  

You may consider returning to search for other equivalents or circumstance causes later by trying to put what you want (or some other equivalent or other circumstance cause) into a form that allows you to apply and use what you know, the given information, and likely causes in some way.

-If necessary‡, multiply by 1 or add 0 in some useful way to do this (method 18).

Notes

† You reason what “usually” happens or causes something by using defeasible reasoning (method 33).

This means you do these steps by considering what “usually happens”, what “usually works”, or what might “seem to work” given your experience solving problems or observing other problems being solved.

Example

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

Sub-Method 2a

Take your problem as solved [7].

Class of Method

A relate sub-method.  A Primary Problem-Solving Strategy.

Use When

When you need another way of finding equivalent(s) to your problem.

How

1.  Determine what properties the hypothetical solution must have by what is implied by the given information and by what is wanted (i.e. the objective of the problem).  Do this by applying the experimentation method (method 7) to search for truths about the hypothetical solution to the problem.  You may also use skeptical defeasible reasoning to consider what the properties are for the hypothetical solution.

2.  Once you know these properties, search for the object that satisfies such properties.

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.

Example

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

Sub-Method 2b

Consider what ideas or statements would be helpful with solving the problem if they were true.

Class of Method

A relate sub-method.  A way of finding “circumstance causes”.

Use When

When the relate method has a need for this method.

How

1.  Begin working backwards (sub-method 2d), but using defeasible reasoning (method 33) to take large steps in the backwards direction.  Consider the set of possible needs that would get you what you want.  Be sure to work backwards as far as you feasibly can.

2.  Experiment with the entities in your problem (method 7) and search for patterns (method 12) to find out which of those needs you could obtain or prove true from the given in the problem.  Do this for those needs that seem most promising to achieve.

3.  Try to prove hypotheses you obtain from this experimentation to use for your problem.

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.

Example

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

Sub-Method 2c

Try to put what you want into a form that allows you to apply and use what you know and use the given information.

Class of Method

A relate sub-method.  A way to find intermediate steps, circumstance causes, or equivalents.  This method can assist with “working backwards”.

Use When

When you have seemingly sufficient resources and you need to find an Intermediate step, Circumstance cause, or an Equivalent.

How

1.   In order to do this, go to method 15(c) replacing "entities" with "what is wanted".

2.  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.

Examples

For example, you want x, but you put it into the form x squared to allow you to use a theorem or formula.

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

Sub-Method 2d

Work backwards [7].

Class of Method

A relate sub-method.  A Primary Problem-Solving Strategy.

Use When

When there aren't many branches of solution paths coming from the backwards direction [Engel pg. 377].

How

1.  Assume that what you are trying to achieve is already done.

2.  Consider what second to last step could have led to the final step.  In order to do this:

-Consider what statements are equivalent to your desired conclusion (in the case of a proof), or solving for what sub-problems or entities would give you your desired result.  For example, if there is a theorem that would give you what you want if you could satisfy the theorem’s condition, then your problem is to satisfy the conditions of the theorem, or consider how you could transform your problem in a way that puts the details of your problem into the necessary framework to apply that theorem.  Also, if what you want has various defining parts, then solving each sub-problem of finding the defining parts equivalent to solving your problem.

In order to do step (2), you may also:

        -Consider what entities, data, etc. if you did have would give you your desired result.

-Consider what statement if it were true or what mathematical situation that held would have likely helped you achieve the last step of solving the problem (often using defeasible reasoning).

-Consider what causes would likely produce such a result or implication [7].  Do this by using skeptical defeasible reasoning on what usually produces similar‡ results, or formulate a useful conjecture that implies the conclusion of the theorem.  Then you experiment with concepts and information related to the desired conclusion.  You experiment (search for truths using method 7) on assumptions about the information in your conclusion to decide what implies your desired conclusion.

-Consider where you have seen a similar‡ final result produced [7].

-Consider what others do to obtain such a final result [7].  (See method 26 for using plans similar to plan in similar problems.)

3.  Then consider which of the possible second to last steps would most likely be achievable from the first step.

4.  Continue moving backwards by restarting this method on the second to last step of solving your problem in trying to find the third to last step that solves your problem and so on.

5.  Eventually you may come to a point in the working backwards process where there are too many possibilities for the next previous step.  You may then need to restart the problem-solving process on the new problem of working forwards using other methods to get to your nth to last step you solved for in the backwards direction.

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

Often you can work the steps of your problems in the backwards direction using defeasible reasoning without filling in the details of each step in the backwards direction yet.  Later you can solve each sub-problem of filling in the details for each step.

‡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.

Example

See example problems 1 and 11 of chapter 6 (Examples of Problem Solving).