Method 14

Simplify the problem.

Class of Method

A Primary Problem-Solving Strategy.

Use When

When the problem has much complicated combinations of data to relate, or complicated combinations of data to show.

How

Use one or more of the following sub-methods according to your best judgment (defeasible reasoning):

Sub-Method I)  Use without loss of generality arguments†.

Do this when special cases of the problem can be solved, and all other cases would seemingly be solved using the exact same method of solution of your special case with a minor alteration.  (This usually follows after noticing a plan that would work for one case of the problem and recognizing that the same type of plan could be used for other similar cases—see method 22 for breaking the problem into cases, or see part (c) of this method).

In order to do this:

1.  Create your new problem as solving the sub-problem that is restricted to the smaller class of special cases.

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.

Sub-Method II)  Exploit symmetry.  (see method 16)

Sub-Method III)  Break the problem into various cases that may be easier to solve separately. (Commonly known as “divide and conquer”.)

-Do this when you can likely find methods of solution for specifications of your problem, and the method of solution for different specifications of your problem would likely require different approaches or methods of solution.

1.  Start the problem-solving process on each of your special cases as a sub-problem.

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.

Notes

† Without loss of generality means that the general case of your problem is solved using the same method as for a special case.  Usually the problem is solved with the statement: “Without loss of generality, (specific problem mentioned).”

Method 15

Put the problem’s information into a form that can be more easily manipulated and controlled.

Class of Method

A Primary Problem-Solving Strategy.

Use When

When the problem’s information is in a form that is difficult to manipulate.

How

-If the “Use When” condition for 15a is satisfied, then go to sub-method 15a.

-If the “Use When” condition for 15b is satisfied, then go to sub-method 15b.

-(15c) Otherwise, try to put entities into forms that allow known theorems, methods or strategies to be applied.  In order to do this:

1.  Consider (or search) for theorems strongly related or similar‡ to entities that are in your problem where such theorems have a high likelihood of helping you to manipulate your problem to get your desired result or sub-result.

2.  Take each such theorem, and consider the possible ways entities of your problem would be represented or transformed in order to apply those theorems to your problem in a useful manner.

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

‡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.

Sub-Method 15a(i)

The invariance principle [40].  Search for entities (auxiliary elements) that do not change but contain the information about complicated changing and transforming entities in the given of the problem.

Use When

When data or information in the problem undergoes consistent changing and transformation in a way that is difficult to manipulate.

How

Use one or more of the following sub-methods by using good judgment (defeasible reasoning):

Sub-Method I)  Search for an invariant that relates directly to what you want.  Accomplish this by doing the following:

1.  Identify the expression(s) that change under some variable or series such as iteration over an index.

2.  Consider with respect to what entity classes‡ you need an expression that is invariant with respect to the subclasses (specifications) of those entity types.  In order to do this, create a list of possible ways you could specify the entities that vary into more specific subclasses.  For example, real numbers can be specified into subclasses as positives, negatives, rationals, irrationals, etc.  Check each of these possible subclass definitions to consider which of these subclass definitions that if you had an invariant† within each such subclass would solve or help you to solve your problem.

3.  Search for a function or expression that does not change†† as the entity (such as a variable or series) changes or iterates within the needed subclasses.  Do this in a way that relates directly to what you want if possible.

You do this by solving the sub-problem of finding the invariant according to the way the data is changing and transforming—you choose a function that takes the changing and would make an aspect of it no longer be changing with respect to what is wanted.  Sometimes the invariant does not need to relate to what is wanted immediately, but you will figure that out later with your given invariant as new information.  Doing this may be too difficult to do, and you may want to go to (Sub-Method II).

Note:  Instead of following the steps above, you may consider solving the sub-problem of finding both the invariant and the subclasses that the invariant is invariant with respect to that would help solve your problem.  Doing this may be less time consuming if there are too many possible separations of your varying entities into subclasses that you would otherwise have to consider.

4.  Then try and relate these auxiliary elements with the parts of your problem to solve your problem.  In order to do this, restart the problem-solving process on your new invariant to relate it to what is wanted in the problem.

5.  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.

Sub-Method II)  If the previous sub-method doesn't provide the needed information to solve your problem, then search for any invariant that likely relates to what you want indirectly.  Accomplish this by doing the following:

1.  Take the difference between the varying entities, or the difference of squares, sum, sum of squares, division of the entities, other functional operations, etc.

2.  Then try to derive an invariant from that expression as your NEW problem to solve using again all problem-solving strategies.

3.  Remember, you may derive an invariant, and continue problem solving using that invariant, but later need to return to this method to find another invariant that relates to a part of the problem-solving process later.

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.

Sub-Method III)

1.  If you already have an expression where its defining parts can vary, then consider what most general specifications of the defining parts would solve your problem directly.

2.  Then consider in what way varying those defining parts would then indirectly give you the solution to your problem due to some invariance of your expression.  (e.g. E7 pg 5 Engel)

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

† An Invariant is a function, expression or some other representation that does not vary with respect to the subclasses of an entity type.  All representations of invariants can be converted into an invariant function.  An invariant is therefore a function whose input is the varying data, and whose output is always the same.  For example, if your invariant is xn > yn for all n (here this is an invariant with respect to relative sizes), then the function f(xn, yn) = xn – yn is the functional representation of the invariant where the f(xn, yn) > 0 (meaning that the function is invariant with respect to it’s sign.

‡ For example, you may need an expression that is invariant with respect the integers mod 5—meaning that the expression is invariant in each such subclass.

†† Here “Does Not Change” means that the auxiliary element you create does not change in some manner with respect to some defining aspect of change.  For example, you may only need your invariant to not change it's sign, or its entity type in some way such being invariant with respect to being rational, or transcendental.  Or for example being invariant with respect to its value mod(n).  When possible, you should consider searching for an invariant in the manner your invariant needs to NOT vary in order for the invariant to be of use to your problem.

The invariance principle is a tool for information simplification (organization) converting varying and difficult to use resources (information) into a usable form in order to control the problem’s information.  The invariant becomes a more convenient resource for attacking the problem.  This principle of organization takes infinitely or otherwise greatly varying data into finitely varying or controllable data. The invariance principle could also be considered as a method of representing information in the problem in states of lesser or minimal entropy.

  Creating useful invariants requires creating the invariants to contain the needed information contained in the varying data.  For example, the constant function (mapping all varying data to a constant) is an invariant, but this does not capture the information about the varying data.  Signs that your invariant likely contains useful information is that the invariant is a closed form function (meaning that it is a function involving standard mathematical operations) that involves your inputs, and the function does vary with inputs that are not restricted to the varying data in your problem.  Your invariants do not necessarily need to contain all of the information in your varying data (meaning that if it did would imply you could reconstruct your varying data perfectly from the information in the invariant).  Your set of invariants simply needs to contain sufficient information from the varying data that is needed to solve the problem.

Other Notes

The invariant is part of the information about what doesn't change about what does change.  Therefore finding invariants is strongly related to finding patterns.

Interestingly, symmetry is a type of invariant.

A key principle to data compression is finding invariants in the data—finding what stays the same and what changes in the data.  Then to find what stays the same about what changes and what changes about what changes in the data, then what stays the same about what changes about what changes and what changes about what changes about what changes, etc.

Examples

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

Other examples include: Energy and Lyapunov functions that never change to a higher level (only decrease), Lasalle's invariance principle, Physics invariance principles.

Sub-Method 15a(ii)

Create an invariant, instead of just finding one.

Use When

When your problem allows great freedom of choosing a specific element of a class that would be used to control or specify some difficult aspect of your problem.  Especially do this if there are commonly known invariants that are used for solving similar problems in the field of mathematics related to your problem.

How

1.  Consider what invariants that if you did have (combined with other representations or steps of solving your problem) would solve your problem.

2.  Solve the sub-problem of considering which type of those invariants you could create given your required restrictions in the problem.

3.  Solve the sub-problem of creating those combinations of needed invariant(s) that are most feasible to create.

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 problems

In control theory stability of a system is often the goal.  This stability can be achieved often by creating a control that makes a certain representation of the dynamics have a Lyapunov function (invariant), and therefore control the dynamics to the desired state.

Sub-Method 15b

Use the extremal principle [40].  Search for an extremal that allows you to show existence of something needed in the problem.

Class of Method

Method for Showing Existence of an Entity.

Use When

When a function can be created with the information in the problem that has extremes in a way that would be useful to help show the existence of an entity whose existence would be useful for solving your problem.  You use the extremal principle to show existence.  The extremal principle is mostly constructive, giving an algorithm to construct mathematical entities.

How

1.  Pick an object, arrangement, geometric figure, or other mathematical entity that maximizes or minimizes a function that would likely be of use to show existence of what you need.  If there are several objects that are extremal, then it usually doesn’t matter which one you use [Engel pg 39-57].  You do this by searching through the relations between entity classes in your problem and choose an extremum from those relations (functions) that would likely be useful.

2.  Solve the sub-problem of showing that the resulting object has or implies a desired property by taking advantage of the fact that a slight perturbation would further vary the given function from its extremum.

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

This method is often used together with proof by contradiction (method 35) by choosing an extremum under the faulty assumptions created when using proof by contradiction.  Then the method is used to show existence of yet another entity more extreme than the original extremum chosen thus obtaining the needed contradiction.

The least upper bound axiom provides an extremal to create completeness of the real numbers and is often used for showing existence.

Example

See example problems 6 and 7 of chapter 6 (Examples of Problem Solving).

Method 16

Take advantage of symmetry [7].  Convert your problem into a problem that allows for useful symmetry.

Class of Method

Both a “Resource” and “Problem Simplification” Method.

Use When

When algebraic or geometric symmetry occurs in your problem and is likely to help solve the problem.

When you may be able to convert your problem into a problem with useful symmetry, and after spending much time searching for resources, your current resources in the problem are still difficult to apply.  (Use this before trying methods 10 or 11.)

How

1.  If you have already found symmetry in your problem, then add the symmetry to your list of resources, and search for the next method that best suites your given information.

2.  If you are searching to convert your problem into an equivalent problem that has symmetry then:

Begin solving the sub-problem of finding a way to convert your data, expressions, or geometric figures (using auxiliary elements or other manipulations) into a form that has a type of symmetry that would likely help solve your problem.

3.  Search for those ways that would be both useful and allow you to convert difficult to work with expressions or diagrams into symmetric information that can be manipulated meaning:

-Mathematical relationships would exist to manipulate, or it is likely that it will be possible to create relationships to usefully manipulate.

-Some mathematical entities are symmetric in some aspect of subsets of variables, expressions or geometric sub-parts.

-There is a high likelihood of the symmetric relationship being useful for the overall goal of the 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.

Notes

Algebraic symmetry occurs when one variable or algebraic sub-expression is interchange-able with other variables or algebraic expressions in some permutation that preserves the original main algebraic expression.

Method 17

Try to put the problem into a form that seems easier to work with; transform the problem into something else you know that looks more convenient to work with.

Use When

When information is cluttered and difficult to juggle in your mind, or extremely inconvenient to work with.

How

1.  Make expressions more convenient to work with by organizing the information or expressions in various ways that may affect your ability to manipulate the expressions and data.  In order to accomplish this:

a)  Organize and de-clutter the information and put it in a format that is easy to read and manipulate.

b)  If there is a lot of clutter in a specific relationship or expression, then try to simplify the relationship or expression.  For example, you may try to convert a large formula by factoring and grouping expressions in certain manners into another representation that would seemingly make the expression more convenient to work with or relate to other parts of the overall problem.

c)  If there is a lot of inter-relational clutter and scattered information, then consider how you could relate that information together to create main, summarized, or simplified relationships that contain all of the information from the previous scattered data and information.  Do this by:

Considering in what way you could combine the scattered information into something useful to your problem by using general problem-solving skills such as sub-method 2d (working backwards), method 12 (search for a pattern), method 33 (using defeasible reasoning on implications of combinations of the scattered data), method 1 (focusing on what you want), trying to fulfill a step of your plan, etc.

2.  Other methods that may help make the problem more convenient to work with include method 15a (search for an invariant), method 11 (Search for a way the problem might be reconstructed under a different type of model), etc.

Be careful not to use method 11 unnecessarily because this could easily be a hopeless cause that would waste your precious time.  If it seems likely after some thought that using method 11 would be a time waster, then choose another method.  If you do decide to use method 11 out of desperation, then do your search for a model without yet having had information in your problem strike you as similar to another branch of mathematics.  Doing this can be complicated and time consuming and hence is why this additional step to method 11 is used when your problem is very difficult.

Notes

Clutter is a burden to the mind and often hinders the problem-solving process.

Note that sometimes introducing carefully chosen forms of complexity into a problem can help you solve some types of problems, and in these situations, over-simplifying the problem could increase the difficulty of solving your problem.

Method 18

Multiply by 1 or add 0 in a way that allows you to manipulate the expression to get what you want.

Class of Method

A Secondary Problem-Solving Method.

Use When

When you need to manipulate an algebraic expression, other methods have proven difficult, and multiplying or adding an expression to your expression would make it easier to manipulate your expression in the way you need.  (Usually you multiply by 1 in a division sub-expression you need to manipulate, and add 0 in a sub-expression that is a sum.)

How

Sub-Method I)  If multiplying by 1, search for a value (α/α) that when algebraically manipulated into your expression allows you to better manipulate the expression in some desired manner.  Accomplish this by doing the following:

1.  Consider† the set of allowed values for (α) on the denominator that would not inhibit your ability to perform the desired manipulation to your expression.

2.  Consider what possible expressions multiplied to the numerator would allow you to manipulate your expression as you desire.

3.  Find a value of (α) that satisfies the first two conditions and choose that value of (α) for use to multiply by (α / α) and algebraically incorporate it into your expression. Of course sometimes you may need both (α) and (1/α) to allow you to manipulate your expression as you need, and you then must choose (α) carefully so that both (α) and (1/α) will allow you to manipulate your expression as needed.

4.  Begin to manipulate your expression according to your needs for solving the problem.

5.  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.

Sub-Method II)  If adding 0, search for a value (α - α) that when algebraically manipulated into your expression allows you to better manipulate your expression.  Accomplish this by doing the following:

1.  Consider the set of expressions for (a) that when added to your expression would allow you to manipulate the expression.

2.  Consider what expressions for (α) when subtracted from the expression would not inhibit your ability to do the desired manipulation to your expression.

3.  Find a value of (α) that satisfies the first two conditions and add (α - α) to algebraically incorporate it into your expression.  Of course sometimes you may need both (α) and (-α) to allow you to manipulate your expression as you need and you then must choose (α) carefully so that both (α) and (-α) will allow you to manipulate your expression as needed.

4.  Begin to manipulate your data according to your needs for solving the problem.

5.  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

†In steps 1 and 2 you may interchange “denominator” and “numerator” depending on where you need an additional expression (α) to manipulate your original expression.

Example

See example problem 12 of chapter 6 (Examples of Problem Solving).

Method 19

Carefully throw away undesirable parts of expressions if possible.

Class of Method

A Secondary Problem-Solving Method.

Use When

When there is a part of your expression that if excluded would allow you to solve the problem.  For example, a square root symbol, the square of something, etc.

How

You have to be careful to justify throwing away an undesired part of an expression.  For example, if you are proving an inequality and part of the expression is on the greater side of the inequality but negative, then you can remove that unhelpful part of the expression (that is if it can’t help you in any way or hinders your progress).

Sometimes method 18 (multiplying by 1 or adding 0) can help you to do this without violating conditions of the problem.

However, often you need to find another problem that if solved would solve your original problem.  For example—does creating a proof about the square of the expression allow you prove something about the expression?  You may need to use methods 1, and 2. (Focus on what you want and work backwards).

-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

Proofs in calculus where there is a square root or square in the way.