Plan Creation and Plan Variation Strategies

Method 24	Create and follow a plan [7].
Class of Method	A Primary Problem-Solving Strategy.
Use When	When many intricate steps to solve the problem seems likely.
How	1. If your problem would seemingly require a non-trivial amount of work to create an easy to solve plan, then first use methods for creating plans that are more promising and less time consuming such as method 26 and method 27 if possible. (These methods use plans based on similar problems that have already been solved). 2. Create a plan by doing one of the following: -Use the relate method (method 2) to search for the simplest and most promising plan that requires the fewest required general steps to solve. You should also consider creating plans by using method 2 that may require more general steps to solve, but are more promising to implement. These plans are called relate plans. There are sometimes several possible paths created by relate plans that can be considered. -Consider plans that involve starting by implementing methods. For example, method 32 (guess and check), or other methods. Then consider what sub-problems and other varieties of methods would need to be applied or be the most appropriate after implementing the initial method you considered. Continue considering step by step which methods and sub-problems would likely seem to best follow after each step until it becomes hard to consider further steps without actually implementing all of the previous steps. These plans are called heuristic plans and paths. Carefully analyze and consider the difficulties, the promises, and the amount of time seemingly needed to implement each of these paths. 3. If you were successful in the previous steps, then go to step (5). 4. If the previous steps were difficult to apply, then depending on the circumstances of the problem ,consider: -Using a form of problem simplification by using simplification methods. -Searching for resources (method 6). -Reformulating the problem under a different model, changing cumbersome notation, or whatever else “seems” necessary to prepare your problem for creating a plan. When you feel that you have enough information to begin searching for a more promising plan, then return to the beginning of this method. 5. Carefully check and possibly experiment or create sub-plans for implementing each step of the plan you created in order to diagnose some degree of possibility or feasibility to implement each of the steps of the plan. If your plan seems unpromising, then save this idea for now and restart this method to search for a more promising plan. 6. Out of the set of plans you have created, consider the plan that seems simplest or overall most feasible to accomplish. (When you try to implement this plan, you may find this plan to be harder than you thought, and you may decide to either create another plan, or choose to implement another one of the possible plans that originally may have seemed less feasible than the first plan you tried to implement). 7. Follow the plan by solving each sub-problem that defines each step of the plan that your defeasible reasoning outlined for you. However, be opportunistic while following your plan—meaning that whenever you happen upon relationships, symmetry, or other information that is likely useful, then search through the key parts of your plan considering how this new information could help to simplify a part of your plan. Also consider if you could use this information to form another plan that is simpler or more promising than your original plan.
Notes	Each of the sub-problems you need to solve to follow your plan may also require the creation of a plan to successfully solve your problem.
Example problem	See example problem 13 of chapter 6 (Examples of Problem Solving).

Method 25	Consider applying commonly used methods to solve classes of steps to problems that you learned during your analysis of problems solved in textbooks on the subject you have read.
Class of Method	Subclass of Defeasible Reasoning.
Use When	When you recognize similarity in aspects of your problem or sub-plan to other problems. Also use this when you are unfamiliar with common strategies in the field of mathematics your problem belongs to. Consider those techniques you have seen before that were used to solve a problem or sub-problem that has entities and relationships that have relationships and entities (maybe not all) that are equivalent in some way to a sub-problem of your problem. Then you should use this method when those techniques would help to solve a part of your problem that would also be useful in getting what you "want", or helpful to get something that would assist you in getting what you want.
How	1. Search for those commonly used methods that would be useful for a plan for your problem. In order to do this, analyze the solutions to problems already solved that have some similar “types” of obstacles and difficulties that are in your problem. 2. Consider how and if a plan can be created to overcome the “types” of obstacles in your problem by taking advantage of the commonly used methods, or some kind of variant of the commonly used methods. 3. Begin to implement the commonly used strategy in your "similar‡" problem by creating the "similar‡" sub-problem that would use the common methods and follow method 26 (Find similar problems and their solutions).
Notes	This process is often used when doing defeasible reasoning to create a plan to solve your problems. ‡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 26	Find similar‡ problems and their solutions. Consider if there is a useful similar method of the similar problem's solution for your problem, or a similar type of plan to apply to solve your problem.
Use When	This is one of the first methods you should implement after method 1 (focus on what you want) because this is the most promising method for a potential quick and easy solution to your problem. Also use this method whenever you opportunistically happen upon solved problems that are similar to your problem.
How	1. Search for problems in publications that have relatively many mathematical entities or many defining parts of entities that are equivalent to those entities in your problem where the majority of the relationships between the corresponding similar entities in your problem and in the published problem are equivalent. Such relationships could also be generalizations or specifications of the relationships in the published "similar" solved problem. 2. Consider the plan of the similar problem, and consider what parts of all steps of the plan could be incorporated into your problem according to the equivalency of the relationships of the defining parts of the entities in your problem and in the example problem. 3. If it seems unlikely that parts of the plan of the similar problem will to be of use to help solve your problem, then restart this method from step 1 to try and apply a modification of another similar problem's solution to your problem, or choose another heuristic (method). 4. For relationships between entities in your problem that are generalizations of relationships in the similar problem on "similar" entities, consider which of the mathematics used to solve each of the sub-results using such relationships can also be used in your problem. If doing this is not possible, then consider in what way you could change the process to allow for the general part of such relationships to be incorporated into your method of solution. 5. If the relationship in your problem is a specification of the relationship in the similar problem, then you are usually safe to apply the same approach to using the relationship in your problem that the similar problem's solution used the relationship. However, doing this may not be possible if the similar problem's use of this relationship incorporated many other relationships together that are not related to your problem. 6. If similar use of a sub-argument needs to be incorporated where many similar relationships are applied together in one part, then you would have to be more careful to make sure that applications of the combinations of relationships in your sub-problem from the similar argument would still apply correctly. 7. If you have trouble with the previous steps, then consider what variation of the parts of the sub-argument for your problem that uses the combinations of relationships would still work. a) You do this by finding the parts of the sub-argument for your problem that do not work. b) Restart the problem-solving process for considering how to make the parts of the sub-argument work. 8. Find all parts of the similar plan that may need significant alteration to solve your problem and make those your new sub-problems. 9. Restart the problem-solving process for each of the new sub-problems. 10. 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.
Example	See example problem 13 of chapter 6 (Examples of Problem Solving)

Method 27	Convert a problem you don't know how to solve into a problem that you do know how to solve.
Class of Method	A Sub-Method of Method 26.
Use When	When your problem has many aspects that are similar to problems already solved.
How	1. Consider the similar solved problem. 2. Consider what alterations to your problem would make your problem essentially equivalent to the similar problem. 3. Consider the alteration to your problem that seems most feasible to accomplish. 4. Use other problem-solving strategies (e.g. method 18, 2, etc) to consider how to accomplish the needed alteration. 5. If finding this alteration proves difficult then go to the next alteration to your problem 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	The reason this method is presented in its specificity is because of its common use.
Example	See example problem 9 of chapter 6 (Examples of Problem Solving).

Method 28

Vary your plan.

Use When

When you have few signs of progress after difficulty implementing your current plan.

How

1. Do a root-cause analysis (method 29) on your initial plan to find which part of your plan is not working properly.

2. Consider if you may be able to prove that it is impossible to follow your plan, hence meaning you may have to think “outside the box” (method 34) on your plan. Quickly try to prove or understand why it may be impossible to follow your plan by solving the sub-problem of showing that the problem cannot be solved with the restrictions to the problem and method of attack that you have assumed.

3. Find a variation of the problem and/or change in plan that would seemingly make your plan work given the cause that hindered your original plan from working.

4. If the previous step did not work, then go to method 24 (create a plan) and create a new plan, or consider using one of the other possible plans you may have considered but not used the last time you used method 24 on your 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.

Method 29	Do a root cause analysis†.
Class of Method	A Method of Diagnosis.
Use When	When there is some unidentified cause(s) to a problem that is hindering an important process.
How	Identifying the Cause(s) (Part I) 1. Specify and define the observed issue(s) whose cause(s) you want to find. You may have several observed issues that affect each other in some complicated manner. There may be multiple root causes to multiple observed issues, there may be one root cause to multiple observed issues, or there may be multiple root cause to one observed issue. Make sure that you properly understand the observed issue(s). Identify for which cases the observed issue(s) are a problem, and for which cases they are not a problem. If time and your situation allow, solve the sub-problem of trying to reliably reproduce the undesired observed issue(s) by making changes to the system that governs the process. Doing this helps you to better understand the observed issue(s). 2. Consider all the possibilities in your plan, physical system, or etc. that could be the immediate cause(s) of the observed issue(s). Note that you may be but are not necessarily considering the potential root cause(s), but the potential immediate cause(s) that would be caused by the root cause(s). In order to do this, list as a hierarchy each primary part of your system, and each sub-part of the primary parts, etc. of your system that could affect your observed issue(s). Each main part of your system is composed of smaller parts that are likely composed of even yet smaller parts. 3. Reduce the number of possible causes to a manageable size through investigation, experimentation, intuition (skeptical defeasible reasoning), or by other problem-solving means. Here you don’t want to check every suspect cause, but create experiments that would eliminate large sets of suspect causes, or to figure out what is the probable cause through defeasible reasoning and what would usually be a cause from the experimental results. Investigation: Sometimes your problem exists in some cases, but not in others. Consider what is the difference and what is the same between what does work and what doesn't work. Then analyze the set of possible explanations for what would most likely cause the differences between cases where the problem does and doesn't surface. Look at the implications of the observed issue(s) to eliminate potential causes or to directly identify the possible causes. For example, a properly functioning sub-system “A” wouldn't allow such observed issues to exist, therefore either sub-system “A” is not working properly, or something that affects sub-system “A” is not working properly. On the other hand, you may have a situation where one sub-system could not affect the observed issue(s), therefore that sub-system is not a cause of the observed problem. Experimentation: Damaging experiments Some systems can be damaged or accumulate more issue(s) by implementing certain experiments. For example, experimenting by trying to start a car when the issue might be a broken timing belt, could damage your engine further if you have an interference engine. When you create experiments, be careful to consider the potential of causing other new issues. If an informative experiment is created where the expected benefits of quickly finding the cause by implementing the experiment outweighs the potential costs of causing other issues, then the experiment is sound. Looking for other issues, and non-issues Investigate to find out what parts of your system do not have issues contributing to the problem(s). List the parts of the system that have issues, and the parts that do not have issues. Knowing what parts do not have issues helps you to now eliminate many potential causes—or at least suspect certain potential causes over others, giving you more guidance for future investigation. Experiment to find or consider what other negative events (issues) could be or are happening. Observing other issues will significantly help to eliminate certain possible causes. This is because the potential causes for one issue intersected with the potential causes of another issue are contained in a much smaller set of potential causes. Of course it is possible that one issue comes from one cause, and the other comes from another, but overall, you have a much better chance of identifying the cause(s) by analyzing other issue(s). When there are numerous suspect causes If there are numerous potential causes, then you might also consider trying to set up experiments that balance the expected time (or cost) to implement the experiments, and the expected quantity of potential causes that the results of the experiment would eliminate. Ideally an experiment would be set up such that the results of the experiment, no matter what the results were, would most likely eliminate as close as possible to half of the potential causes. The idea is to cut the possibilities in as close to in half as possible, then cut the remaining possibilities in half again and again until the set of possible causes is small enough to be manageable. However, designing optimal experiments that best balance the cost to implement the experiment with the risks for creating other issues together with trying to obtain as much information about your cause(s) as possible can be very complicated. So you experiment the best you can in a way that balances such important principles—usually trying those experiments that have no risk of creating new issues. When experimenting, sometimes it is useful to consider changing other parts of your system to intentionally cause problems to see what changes in the undesired observation. This is done when your experiment would either help eliminate other suspect causes or directly point to one specific cause as the culprit. Also use skeptical defeasible reasoning to consider what causes such observed issue(s) in similar situations. Use guess and check (method 32) Use method 32 in order to guess the solution to change unverified but likely suspect cause(s). If you fixed the observed issue(s), then solve the sub-problem of creating a theory that would explain what the true cause(s) were and why your guess was the solution. You need to create a theory that explains everything you observe. Having any unexplained issue(s) means that you have not found the true solution, or that there is more than one cause of the observed issue(s). 4. Rank each of the suspect cause(s) in greatest likelihood of being the cause(s). Then rank each of the suspected cause(s) by the expected time it would take to check if the suspected cause(s) are causing the observed issue(s). Choose to check (in step 5) the suspect cause(s) that best balance the least time to check the cause(s), and the likelihood of the suspect cause(s) of being the actual cause(s). For example, you don’t want to completely disassemble a generator to check a potential cause when the cause could be a short circuit in a wire that is much easier to check. Or if you are debugging computer code, you might not want to do a sophisticated re-analysis of the algorithm until you have checked to see if the syntax in that section of code is correct. Checking here is a bit different from experimenting as was done in step 3. This is because checking is a specific “experiment”, or investigation that can only check one possible cause at a time instead of setting up experiments that give potential information about the likelihood of a variety of potential causes as was done in step (3). 5. For those suspect cause(s) chosen in step (4) that cannot be caused by something else, check to find out if those suspect cause(s) truly are the cause of the observed issue(s) and go to (Part II) for any of those cause(s) that are found to be true cause(s). If after checking, you find that those suspect cause(s) could not be cause(s), then use good judgment to decide weather to go to step 3 or 4 to search for their next most promising cause(s) to check. What if step (2) revealed that your currently suspected cause is the cause? For those suspect cause(s) that can be caused by something else go to step 6. 6. After checking (if checking is feasible) to be sure that your suspect causes are truly being affected negatively in some way, restart the process of searching for the root cause(s) of the primary suspect cause(s) to find its cause(s) by restarting this method on the suspect cause(s) as the new observed issue(s). You may have found the most promising potential cause(s) of the observed issue(s). However, those cause(s) most likely were caused by yet another cause. This is why you restart this method within this method (recursion) on finding the cause(s) of the suspected cause(s) (those suspected cause(s) chosen from step 4). Fixing All Identified Causes (Part II) 1. Begin other problem-solving strategies to fix the root cause(s) as best as possible without creating other new issues. If it is not possible to avoid creating other issues, then fix the root cause(s) by creating only the new issues that either you could most likely fix, or the new issues would be more acceptable than the current observed issue(s). 2. Check to make sure that there are no other issues after fixing the root cause(s). If there are more issues, then restart this method to find their root cause(s). 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	†Root cause analysis is a problem-solving skill that involves searching for what prevents something from working properly. Examples of using this skill includes figuring out why a plan to solve a math problem doesn’t work, debugging computer code, diagnosing car problems, understanding why other types of plans don’t work, etc. Common Error to Avoid: When doing a root-cause analysis, you can make the mistake of thinking you have found the root cause, but what you have really found is another symptom of the root cause. If you begin to repair a symptom, then the root cause still remains, and you will likely create even more issues by patching up the symptom. In order to avoid making this mistake, be sure that when you think you have found the root cause, to carefully consider if there could be other cause(s) to the observed symptom that may be the cause, or may in fact be caused by other issue(s). For example, the bent frame of a car could cause a wheel to shift backwards, and a repairman may think to just shift the wheel forwards. The root problem was that the frame of the car was bent, but was never repaired, and now the wheel is offset as well creating even more issues causing problems for the vehicle. Similarly, finding a symptom of the difficulty in creating a proof can lead you to try to solve the issue of a symptom of the difficulty causing you to never address the main mathematical difficulty to creating a proof. Don’t get caught in a rut assuming that certain parts of you system could never be part of the cause unless you have logically eliminated its possibility. For example, once when I was doing a root cause analysis, I had identified the primary system of the cause of the observed issue. I checked the details of that cause several times over in order to fix those details meanwhile assuming that nothing else affected that cause. It turned out that something else I had previously assumed was perfectly correct was the root cause that caused the cause of my observed issue. But I had been caught in a rut repeatedly checking something else that I knew was not the root cause! I know someone that completely disassembled the fuel pump of a car because no fuel was being injected into the car’s engine. It turns out that the true problem was that the car was out of gas! Mistakes like this come from either failing to check the more convenient to check potential causes before investigating the more costly to check potential causes, or not carefully considering and listing potential causes before investigating potential causes. Sometimes the root cause may be making something else actually work in some twisted way, and make you ignore problems that exist which may affect the root cause analysis all together. When something works in some twisted wrong way, then that usually means that it actually isn't working and that another issue actually made you think it is semi-functional.
Example	Examples are problems from people whose profession depends significantly on doing root cause analyses: mathematicians, scientists, problem solvers, teachers, business analysts, mechanics, doctors, nurses, code debuggers, psychologists, technicians, etc. Anything that requires diagnosing involves doing a root cause analysis.

Method 30	Break down an entity of the problem into its defining parts and resume problem-solving using its defining parts.
Use When	When there are few or no theorems, axioms, or methods that relate to that defining entity in a way that is likely useful to your problem. Use this method also when theorems, axioms, and methods that do relate to that entity have proven difficult to apply to your problem.
How	The title of this method is self explanatory on how to use this method.
Notes	This relates to plan variation, because you change your plan when you have difficulty finding theorems, axioms, etc to an entity in your problem.

Method 31	Break down defining parts of entities in the problem and recombine the parts in some useful manner.
Class of Method	A Sub-Method of Method 30.
Use When	When there are few or no theorems, axioms, or methods that relate to that defining entity in a way that is likely useful to your problem. Or When theorems, axioms, and methods that do relate to that entity have proven difficult to apply to your problem. Use this heuristic if method 30 was not very helpful or if you have many defining parts that can be easily related to help implement your plan.
How	1. Break down the entities that have been difficult to relate in solving your problem into those entities' defining parts. 2. Consider in what way you could recombine the defining parts into something useful to your problem by using general problem-solving skills such as sub-method 2d (working backwards), method 7 (experimenting on implications of combinations of the scattered data to find useful truths), 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. When you do step (2), often you may concentrate on a detail that strikes you. Then on another detail. Then concentrate on various combinations of details and what they together might imply or help you to solve [7]. Try to see new meaning in each detail, some new interpretation of the whole. Try to recombine certain elements to give you another needed truth to help you solve the problem. This process is done by experimentation (method 7). Always keep in mind how you will relate what you decompose to what you want to solve. 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	If you find it necessary to break parts of the problem into their defining parts, then you may still have difficulty using those parts and you may have to break those decomposed parts down into their defining parts. You may have to keep digging deeper if still no useful theorems surface. Instead of applying theorems directly to your problem, you would then likely have to search for theorems (lemmas to prove) that could help you achieve those certain steps of solving the problem.
Example	See example problem 1 of chapter 6 (Examples of Problem Solving).

Method 32	Guess and check (Trial and error): Guess the solution to the problem and check to see how close you are to the solution. Then update your next guess based upon what you learned from your previous guess.
Use When	When you could eliminate most possibilities of solution (or restrict solutions to a smaller range of solutions) to restrict the possible solutions gradually to smaller sets in a way that seemingly would allow you to converge to a solution.
How	Sub-Method I) 1. Eliminate possibilities for potential solutions to the problem by experimenting and considering what restrictions the solution must have. Do this by specifying entities and observing when such specification contradicts possible solutions, or eliminate possibilities by using theorems that could restrict the possible class of potential solutions. 2. Guess the solution as one of the most probable solutions from the set of possibilities that experimental evidence or defeasible reasoning suggests is the solution, or would give you the most information on the restrictions of possible solutions. 3. Continue experimenting using method 7, or use skeptical defeasible reasoning to narrow your possible solutions. 4. Restrict the set of possible guesses by eliminating possibilities based upon the information you obtained from your current guess. 5. Go to step (2) and continue until you converge sufficiently close to your actual desired solution. 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. Sub-Method II) (More advanced approach) Your goal here is to solve the sub-problem of creating a method that makes a guess of the problem’s solution whose information is used in that method to assist in a manner that makes the subsequent guesses made by your method converge to the solution of the problem. 1. Solve the sub-problem of finding an initial guess to the solution in a way such that you can use the information from each previous guess to obtain a new guess closer to the solution such that guesses will converge sufficiently fast to the actual solution of the problem. (This is often implemented with a computer program.) 2. Solve the sub-problem of making sure your final guess is sufficiently close to the actual solution of your problem. 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	Most numerical methods depend greatly on this method. Newton’s method is a classic example of this method. Other sophisticated algorithms that greatly depend on this method include interior point methods, genetic algorithms, simulated annealing, and reinforcement learning. Another example of using this method is finding local extrema of continuous functions by identifying critical points (where the derivative is zero or does not exist). This is because all local extrema will be at critical points. Therefore it is a good idea to try all critical points and check to see which ones really are local extrema. The key to using this method is creating a way to best eliminate possibilities by using theorems to extract as much information as possible with each best chosen guess you pick.
Example	See example problem 3 of chapter 6 (Examples of Problem Solving).

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