The Thorough Problem-Solving Method (Algorithm)
Getting to this point means that your problem is hard. If you are applying this method to solve a sub-problem to another main problem, then you may want to consider implementing a different plan on your main problem to avoid having to solve a hard sub-problem.
To implement this method:
a) Examine other special cases that are at the heart of what made earlier strategies fail.
b) You may also decide to do a root cause analysis (method 29) on the cause of difficulty for implementing previous plans or strategies. For example, notation or the modeling of the problem is sometimes the cause of difficulty in solving a problem effectively.
Failure is a step towards success, and neglecting to properly analyze the root cause of that failure could prevent you from obtaining powerful information.
If necessary, consider a different model of the problem that could fix the cause(s) of difficulty.
If necessary, consider different notation that could fix the difficulty above. Choose a representation of the problem that simplifies the mathematics and the problem-solving process. If you later find that your representation or notation of the problem clutters the problem-solving process, then search for new representations and notations that simplify the problem-solving process [41].
If necessary, consider in what way to “Think Outside the Box” (method 34) that takes into account the hard part of the problem.
If you found something that fixes the difficulty of solving the problem, then begin to implement the plan that would now work. Otherwise:
If at any time during this process you find information that would make a strategy seem very promising, then pause this step to try out the strategy.
Begin to read textbooks and other sources to find the following:
a) Equivalent Problems and Statements—other statements that are mathematical equivalents to solving your problem. For example, equivalent mathematical forms of what you want to solve or show.
b) Resources. Search for more resources that likely relate to your problem and parts of your problem. Later you may do a full study on the mathematical subject if your current search for resources fails to help you succeed in solving your problem. You may find it necessary to do this research now. Whenever you find other useful resources, try to incorporate them into your problem by further implementing the relate routine (method 2) from what you have already done before when you used method 2. For those parts of the problem that you have difficulty finding sufficient resources for:
Create artificial resources for parts of the problem that have few resources to work with, or existing resources for those parts of the problem are hard to incorporate. In order to do this, consider which of the following would be most appropriate:
-Search for auxiliary elements. (method 13)
-Search for advanced symmetry transformations of the problem or parts of the problem. (method 16)
-Use the extremal principle to show the existence of a mathematical entity whose existence would assist in solving the problem. (method 15b)
-Use the invariance principle to put hard to control resources into another more controllable resource. (method 15a)
- If there are seemingly insufficient useful resources for some of the parts in the problem, then do step 2 of method 6 (or method 31) in order to break down entities in the problem that have seemingly insufficient resources for use of these entities. If after breaking down parts of the problem you later discover that there are still seemingly insufficient useful resources for applying some of those parts, then you may have to break down those defining parts even further to their defining parts, etc.
Then search for resources that relate to those defining parts. If later you find no resources for those parts, then you will have to experiment and create resources (theorems, lemmas, etc.) for those parts by using method 7 (experimentation method), and method 12</span> (search for patterns).
Searching for plans and potential problem-solving paths
During this entire process, you should always be monitoring for an opportunistic change of plan. When solving another sub-problem in another path or plan, you may discover something that could make another plan work, or create a better plan by altering a plan currently under consideration. You should have “marker neurons” pay attention to what you discover so that when the neuron is activated, it considers if that previous plan has now become more promising.
1. Each method or approach to solving a problem usually involves solving other sub-problems or other methods which require using even other methods and sub-problems, and so on. Thus using a particular method is the beginning of a large variety of paths to take for trying to find a solution to your problem. Before starting a path (technically the beginning of potentially several paths), you should consider what the result of using each given method would likely be if successful, and consider what the next likely step or method would be after the implementation. Consider the variety of steps that would follow in order to consider the expected complication that would follow the use of a method, and how promising the use of the method and its following steps would be for solving the problem.
††Each of the following is the beginning of some possible paths (plans) alongside and possibly also within other plans you create:
Consider use of the following methods as the beginning of a path to solve your problem:
If necessary‡
a) Carefully throw away undesirable parts of expressions if possible. (method 19)
b) Put information in a more controllable form. (method 15)
c) Use the invariance principle. (method 15a)
d) Use the extremal principle. (method 15b)
e) Multiply by 1 or add 0 in a way that allows you to manipulate the data to get what you want. (method 18)
f) Search for useful auxiliary elements. (method 13) This is usually done for those parts of the problem that you have sufficient resources for, but that are hard to incorporate towards a solution. Should you do this before decomposing those parts of the problem?
g) Search for more advanced symmetry transformations of parts of the problem. (method 16) This is usually done for those parts of the problem where you have resources, but that are hard to incorporate towards a solution. Should you do this before decomposing those parts of the problem?
h) Solve your problem by solving all of its variety of possible cases‡‡. (method 14 sub-method III)
j) Use the more advanced guess and check method. (method 32)
k) Search for problems in textbooks that are “similar” to your problem that you didn’t take the time to search for in the previous faster problem-solving algorithm and consider their plan (method 26). If you do find a similar problem, consider using method 27 convert to a problem you do know how to solve.
m) Taking into account new resources and information found during this process, use method 25—use commonly used methods in that branch that you do not yet know. Investigate commonly used methods that apply to your type of problem by searching for those problems that have similar issues and how they overcome the “types” of obstacles involved in your problem.
n) Consider what special case of your problem that if solved could be used as a main part of a plan for solving your original general problem, or an important sub-problem. (method 22b)
p) Also for certain special types of problems consider other methods like the pigeonhole principle, methods of showing existence, proof by contradiction, proof by contra-positive, etc.
In addition to these potential paths, also consider creating plans in step 2:
2. Try to create a variety of promising plans (method 24). You create a variety of plans because you want to check for simpler strategies before implementing the first plan or path you happen upon. Here you more thoroughly search for plans unlike what is done in the less thorough methods.
3. If all of the plans found so far are expected to be too time consuming, then return to step 2 and keep searching for more plans to create a variety of better plans. Keep searching until searching for more plans is expected to cause more time waste than simply beginning to implement the current most promising plan. If searching for more plans seems too time consuming, then go to step 4. In order to properly balance your time in searching for more plans:
a) Search for another plan until a fraction of the expected time to implement an existing plan has passed. If sufficient time has passed, then go to (c). But if you find another reasonable plan, then go to (b).
b) If you found another plan in (a), then implement that plan to some extent to understand the complexities involved in the plan, and to examine what is needed to implement the plan. Apply defeasible reasoning to do this examination of your plan and try to “see” how promising the plan is compared to other plans you have available.
c) Based upon the information you obtained in previous steps, use good judgment to decide weather to return to (a) and search for another plan, or to begin the next method of implementing your currently most promising plan. If a variety of people are working on this problem (like a research group, graduate students, or a computer) then you can parallelize‡‡‡ this step by assigning plans or paths to each member of the group, and report on progress, time used, and details of progress for the group coordinator to decide the next approach to take on the problem.
Note: in searching for plans, new promising plans may just be variants of already existing plans. You may search for other plans based upon variants of these.
4. Implement the most promising plan or heuristic out of your set of potential paths that seemingly best balances time and probability of success. You apply the most promising heuristic or plan by going to the next method: Implementing your plan. Later you may discover that a path you chose was less promising and you will need to decide whether to search for more plans, or try implementing another plan or path.
Implementing your plan
1. Begin to implement your most promising plan or problem-solving path. Always begin work on the part of the plan that seems most difficult to implement. Doing this is important because if the difficult part of the problem is found to be too difficult, then you wouldn't have wasted time on easier parts of the plan you would no longer need. While implementing this plan or path, if significant time∆ passes, then check your signs of progress (how?) for deciding if you should change your path (or plan), search for more paths (or plans), alter your plan, or vary the problem as explained in the following steps:
a) If you have signs of progress, and your current plan still seems more promising than another path considered earlier and still seems more promising than the expected benefits from searching for more plans, then continue implementing your plan. Be sure to occasionally checking your signs of progress. Otherwise, keep note of what difficulty made implementing your plan difficult. Keeping note of these difficulties is important because the cause of difficulty in one plan may be the key to implementing or creating a successful plan.†
If you decided that your current plan is un-promising, then continue to (b).
b) Put this now more difficult to implement plan into your set of future “potential” plans to implement. Doing this is important because this difficult to implement plan may later become seemingly more promising compared to other plans when investigating the other potential plans.
c) Out of the set of potential paths and plans you created earlier, consider if your next most promising plan or path is not seemingly too time consuming to begin implementing that plan. If it seems relatively promising, then return to step (1) to implement the plan. Otherwise:
If searching for more plans does not seem too difficult, then return to step (1) of Searching for plans and potential problem-solving paths to begin searching for more plans.
∆ A significant amount of time has passed if a specific fraction of the expected time to solve the problem using the next best strategy has passed. These next best strategies are strategies such as searching for other plans, implementing another plan, etc.
If searching for more plans seems too time consuming to do, then:
2. When there are no promising plans to implement and searching for more plans seems to be too time consuming:
a) If necessary‡, more thoroughly search for problems that are “similar” to your problem and consider their plan (method 26). If you find similar problem(s) and their solution(s), then consider using method 27—convert your problem into the problem you do know how to solve. If doing this seems to fail at helping to solve the problem or require too much time to implement (this is relative to how many times you have been to this step), then:
b) Using the information found from the difficulties you encountered when trying to create a promising plan, use method 21 to find out more about the hard part of your problem. In fact you may even redo steps 1, 2 and 3 of the main method to investigate the difficulty of the problem taking into account new information from approaches you have tried since the last time you implemented those steps. Further investigating the hard part of your problem may help guide you to decide where to search for more resources, how to vary your current plans, use other methods, or implement other strategies. Doing this will then likely help you to create more promising plans that take into account the hard part of the problem.
c) Given what you found in (b), use good judgment to decide which of the following method(s) to do:
-Problem variation methods. If you do this, then restart problem solving on the variation of the problem always looking for clues for how its solution can relate to the solution of your original problem. Using this strategy usually helps you to get ideas for plans by solving a variation of your problem. This may also help you find how to get around hard parts of your original problem. Keep in mind the main difficulties and issues that made implementing other plans problematic to implement, and how those difficulties might assist you to implement your current plan.
-Plan variation strategies. This is usually done if you already have some plans, and you were able to identify what was hard about implementing your original plans, and you may be able to figure out a way to adapt to such difficulties.
-Think outside the box according to a proper balance of the increased difficulty of the problem caused by thinking outside the box, and the help that doing so would allow for more plans to potentially solve the problem. Then reconsider paths considered earlier, and how promising those paths are after thinking outside the box. Also consider how you could vary your plans given the changes in assumptions due to thinking outside the box.
Given the information or variety of changes obtained from this step, return to step 1 to implement your current most promising plan.
Notes on the Thorough Method
† An aspect of failure in one plan may be the key to implementing another plan. In varying your plan, or returning to a new plan, the issue that may cause one plan to fail may be the key to implementing a previous plan that you had abandoned. In fact, this is exactly what happened to Andrew Wiles in his proof of Fermat’s Last Theorem [50]. A careful analysis of the aspect that creates your failure in one plan is often the key to solving the problem in a different plan. Again, failure is often a prerequisite to success.
‡ Using the mentioned method is necessary if the conditions are satisfied in the box labeled “when” for the method as written in the method description in the catalog of methods.
‡‡ An example of possible problem-solving paths is breaking you problem into numerous cases and solving each case (as was done to prove the 4-color problem). Searching for hard to create resources is a potential path such as creating lemmas, theorems that relate to parts of your problem. You need to “see” what is around you in your problem by understanding truths that effect parts of your problem.
†† Consider implementing faster forms of a method before trying a more sophisticated form of it. Also, deciding to use these common strategies may require certain resources that you should search for immediately, applying certain methods (heuristics), or solving certain sub-problems involved in each method. Do so as you see fit in order to use such common strategies.
‡‡‡ Problem solving is parallelizable. If you are solving a problem in parallel such as in a research group, then you search for possible general paths† or plans. Then you assign to each member of the group a separate path or plans that best balances how promising the path is versus the time it would seemingly require to implement the path or plan. Doing this is parallelizing. †A path consists of heuristics to use, its associated sub-problems, heuristics for each of those sub problems, sub-sub problems and so on.
Other Notes
The overall problem-solving method should be self-similar for sub-problems (where you start from the super fast method), except that you temporarily abandon sub-problems if your current path becomes less promising than another path or plan. Of course you should save and remember what was done on the path you abandon in case you come back to that strategy later, or parts of that plan become useful in some other way.
Have a “neuron marker” on each plan for just in case you opportunistically happen upon information that would make one of the other plans be promising enough to change your strategy and begin implementing the more promising plan.
Whenever new pertinent or useful information is found and you return to use heuristics, you may find it useful to return to a heuristic you previously used whether that heuristic was useful before or not as your new information may make that heuristic now become a promising approach.
After Ultimate Failure to Solve a Problem
Failure is a common event in problem solving. It is important to know what to do after such a common event in order to adapt to the consequences of leaving a problem unsolved, and also to learn from the failure.
If you have ultimately failed to solve your main problem, then go to the problems solution key if one is available and analyze what you did wrong in your approach to solve the problem, and how to change the way you implement the problem-solving method in the future. Be sure to remember any new methods for solving the type of problem you failed to solve. Failure is a step towards future success if you take advantage of the benefits that failure can give you.
If no solution key is available, and your problem is essential to an important application, then consider how you can adapt and deal with the consequences of leaving the problem unsolved. Consider what other problems that if you solved would lessen the severity of such negative consequences, and try to solve the most seemingly feasible of such problems.
Routine to Follow After Successfully Solving a Problem
1. Review your method of solution.
a) Check each step to your method of solution afterwards to verify that each step is correct.
b) See the solution in parts and as a whole to organize the ideas you used in your mind.
c) Try to see the solution to your problem intuitively in order to strengthen your intuition for bettering your defeasible reasoning. This then helps when attempting to solve future problems that have similarities to the one just solved.
2. Search for applications of the solved problem.
a) Look for possible implications and special cases of the solution to the problem.
b) Consider what application the solution to the problem has.
c) Consider how different specifications of the problem could be applied.
3. Consider how the solution of your problem can improve your problem-solving skills elsewhere.
a) Consider how your method of solution could be applied to solve other problems.
b) Put any learned techniques, methods, or trick in your “toolbox” list of tricks.
4. Vary the problem you solved, and vary the strategy you used to solve the problem.
Try to solve those varied problems and try following other plans to solve the original problem. Do this because your method of solution likely has elements of it that would be key elements to solving similar problems and you could discover useful mathematical principles. Problems come in clusters of similarity that are usually solved quite similarly. [7]
a) Try solving your solved problem more than one way to learn new methods and tricks to help solve similar problems. You may discover new solutions to similar useful problems.
b) Consider possibly generalizing your solved problem and if possible, try finding its solution.
c) Vary the problem for another similar problem that you may be able to use the same method of solution used on your solved problem.
d) Look for a closely related problem in analogy, specification, or generalization. Try to solve the simpler related problems first.
5. Look back at all you did during the problem-solving process to analyze what worked and what did not work and why. Do this in order to recognize where you wasted time, what you did to be less efficient, and how you can be more efficient and effective next time you solve problems.
Some of those strategies that were difficult and you chose to abandon, may have been the best strategy given what was available at that point of the process and you were not prepared for handling such difficulties. Investigate such difficulties in order to better know how to deal with similar difficulties in the future.
Summary of the Main Problem-Solving Method.h
The Extremely Fast Problem-Solving Method (Summary)
1. Understand the problem.
2. Implement any methods you have learned to solve problems of its class. Otherwise:
Somewhat Fast Problem-Solving Method (Summary)
Pre-Process
1. Double check your understanding of the problem.
2. Consider examining special cases of the problem (method 22a) to give you simple ideas about the problem.
3. Go to the problem-classification method.
4. Think inside the box.
5. If necessary‡, simplify the problem (method 14), or reformulate the problem.
The Main Process of the Somewhat Fast Method (Summary)
1. Apply the non-time consuming options from resource and relate methods as seen appropriately.
2. Apply method 25—use common strategies and tools.
3. If necessary‡, consider problems immediately available to you that are “similar” to your problem and consider their plan (method 26). If you find a similar solved problem, consider also using method 27—convert your problem into a problem you do know how to solve.
4. Quickly try to formulate a variety of promising plans (method 24).
5. If you have no immediately available promising plan, then consider breaking down those parts of the problem that seem to significantly complicate the problem.
-Consider thinking outside the box.
-Consider breaking down those entities that you have little resources for, create resources for them, or create artificial resources for them.
Restart this method from the necessary step.
The Thorough Problem-Solving Method (Summary)
1. Try to find the hard part of the problem.
2. Use techniques to analyze the difficult part of the problem and to analyze un-promising plans you likely tried to implement earlier.
-Examine other special cases that are at the heart of what made earlier strategies fail.
-You may also decide to do a root cause analysis (method 29) on the cause of difficulty.
3. If necessary, consider another reformulation or simplification of the problem. If you found something that fixes the difficulty of solving the problem, then begin to implement the plan that would now work. Otherwise:
4. Do a more thorough search for information which you skipped investigating during the previous fast method.
Searching for plans and potential problem-solving paths (Summary)
1. Consider your potential paths, and intuitively follow those paths to consider the difficulty of each.
2. Try to create a variety of promising plans (method 24).
3. If all the plans found so far are expected to be too time consuming, then return to step 2 and keep searching for more plans to create a variety of better plans.
4. Implement the most promising plan or heuristic out of your set of potential paths that seemingly best balances time and probability of success. Consider searching for better plans or implementing the next best plan if you have few signs of progress when implementing your current plan.