Somewhat Fast Problem-Solving Method

Pre-Process

1. Solidify and double check your understanding of the problem.  If necessary‡, create a diagram and create proper notation (methods 4 and 5).  Double checking your understanding is important because often you may unsuccessfully solve a problem due to initially interpreting the problem incorrectly.  If you did misinterpret the problem, then double check your interpretation again, and restart the extremely fast method on the correct interpretation of the problem.

2. If necessary‡, use the quick version of examining special cases of the problem (method 22a) to give you simple ideas about the problem, to understand the problem better, and to get an idea as to how promising certain paths or heuristics would be for solving the problem.

3. Go to the problem-classification method above. (See ***)  You do this after steps 1 and 2 because those steps will assist with helping you to better classify the properties of the problem which will then help you make proper problem-solving decisions.  You may feel (by defeasible reasoning) that you have an insufficient understanding of the branch of mathematics your problem involves, or you may feel that you have insufficient resources to solve the problem.  If so, then you need to use the resource routine (method 6) to learn more about your problem and the branch of mathematics involved until you can classify your situation as having enough of an understanding and resources to begin successfully attacking the problem.

4. Carefully consider what to assume about the solution of the problem, and what approaches to avoid for the moment towards solving the problem.  Here you are structuring your beliefs about the problem in order to focus the next processes on strategies and assumptions that are most likely effective and true.  Only make an assumption that would seemingly simplify the problem-solving process and does not create too much risk of assuming something false or significantly seemingly inhibit powerful or promising strategies to solving the problem.  Doing this is called “Thinking inside the box”.  Thinking inside the box is important because usually your pre-conceptions of a problem tend to be correct.  You can later “Think outside the box” if you are unsuccessful at solving the problem while “Thinking inside the box”. 

5. If necessary‡, simplify the problem (method 14), or reformulate the problem.  If the information in the problem is scattered and it is difficult to juggle the information in your mind, then use method 17 to organize the information in the problem.

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.   Add any alternative restructurings of the problem into your list of “known information”.

6. If the problem has an overwhelmingly obvious need for specific methods (heuristics), then go to those method(s) first before using the following standard steps.

 Some steps of this pre-process are modified from the method given in Schoenfeld pg 111 (analysis stage)

The Main Process of the Somewhat Fast Method

1. Do a quick† “focus on what you want” routine by considering and listing potentially useful equivalents to solving your problem that are not time consuming to look up in a textbook or analyze.

2. Do a quick† resource routine to have clarity for what you are working with.  In order to do this:

a)  Un-thoroughly‡‡ (meaning searching for those convenient to analyze pieces) do step 1 of method 6:  Search for theorems or other known information that relates to what you want without breaking down what you want into its defining parts quite yet.

b)  Search for immediately recognizable symmetries in the problem.  If your current set of found resources causes the classification method to tell you that you most likely have sufficient to relate what you know to what you want, then continue to step 3, otherwise continue to (c).

c)   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 their use in the problem.  Note:  If after breaking down parts of the problem you later discover that there are still seemingly insufficient useful resources for some of those parts, then you may have to break down those defining parts even further to their defining parts, etc.

d)   If you did part (c), then restart††† this method from step 1 of the pre-process if you broke down the entity that is wanted (the desired result of the problem).  You do this because you want to try and solve the problem with a new version of what you want to solve.  Otherwise, decide whether it would be appropriate to restart from step 3 of the pre-process, or to simply restart this step (step 2) and search for resources related to these broken down pieces of the problem.  You need to restart from the appropriate point of the process because you have an altered problem after breaking down entities of the problem.

3. Use strategies to Relate what you Know to what you Want (method 2).

When using the relate method, you will likely need to work backwards (using sub-method 2d) on each of the equivalents to solving your problem found in step 1. You are doing this given your new resources being now more thorough in the approach you started in the super fast approach.

4. If step 3 doesn’t quickly provide seemingly promising results, then temporarily abandon your current plan.  Leave what you learned so far about this plan a “neuron marker”†† just in case for future discoveries that may cause you to opportunistically restart this current plan being abandoned.

5. Use common strategies and tools that you already know from the class, branch or sub-branch of mathematics that your problem falls in (method 25) only if you seemingly have sufficient experience solving problems using such strategies.  If this step fails to help you solve the problem, then:

Deciding to use such common strategies may require certain resources that you should search for immediately, applying certain methods (heuristics), or solving certain sub-problems.  Do so as you see fit in order to use such common strategies.

6. If necessary‡, consider solved problems that are immediately available to you and are “similar” to your problem, and consider using an adaptation of the plan that solved them to solve your problem (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.  If doing this seems to fail at helping to solve the problem or requires a significant amount of time to implement, then:

7. Quickly try to formulate a variety of promising plans (method 24) until you find a relatively easy and promising plan to implement.  Be sure to take into account your variety of resources, how much you were able to relate parts of the problem in the relate step, and the variety of equivalents to solving the problem.  You can try to create different plans that focus on the different various equivalents you found in step (1).

If necessary‡, among the plans you are considering, also consider applying method 32 (guess and check—using the simpler version of the method as outlined in method 32).

Implement your most promising plan.  If your plan becomes difficult to implement, then begin to implement your seemingly next most promising plans.  If you run out of promising plans, then go to step 8.

The following step (step 8) should only be done if you haven’t found good plans using the parts of the problem before you break those parts down into their defining pieces. 

8. If you have no immediately available promising plan, then consider those parts of the problem that you have insufficient resources to apply to those parts for solving the problem.  If breaking down those parts of the problem seems to significantly complicate the problem, then redo step 2 with the exception that you are more thorough in searching for resources that relate to the parts of the problem that are in question.  You may instead even consider searching for “artificial resources” such as invariants (method 15a), auxiliary elements (method 13), extremal elements (method 15b), etc. Then continue the rest of the steps here in the “somewhat fast method”.

‡‡If breaking down parts of the problem that you have insufficient resources to apply to them does not seem to complicate the problem, then if possible, break down those parts of the problem (method 30), and recombine those new parts in a useful manner (method 31).

It may not be possible to break down the parts of the problem in question because you may have previously broken down those parts into their defining parts, and each of their sub-parts into their defining (and so on) until no further breakdown is possible.  If so, then instead:

a)  Consider if the reason for failure to quickly solve the problem was due to your pre-conceptions of the problem from “thinking inside the box” in step (6) of the pre-process and begin to think outside the box (method 34).

If the removal of any more pre-conceptions of the problem would significantly complicate potential approaches to solving the problem, and breaking down parts is no longer a potentially efficient option to solving the problem, then implement the thorough problem-solving method (below after the notes).  Be sure to set a “neuron-marker”†† for strategies you tried to implement up until now.

b)  Redo the parts of this method (the somewhat fast method) that may be affected by taking into account removal of those pre-conceptions that you could reasonably remove without significantly complicating potential approaches to solving the problem.

9. Then restart††† this method from step 1 of the pre-process if you broke down the entity that is wanted (the desired result of the problem).  You do this because you want to try and solve the problem with a new version of what you want to solve.  Otherwise, decide whether it would be appropriate to restart this method from step 3 of the pre-process, or to simply restart the process from step 2 of the beginning process and search for resources related to these broken down pieces.  You need to restart from the appropriate point of the process because you have an altered problem after breaking down entities of the problem.

† The first time through, you do a “quick” and un-thorough process because although you want a method that can solve both easy and hard problems, you also don’t want to use an extremely time intensive/complicated strategy for a problem until you find that a problem is difficult to solve.  In fact most problems will not be very difficult to solve, and implementing a labor intensive strategy at the beginning of every problem will waste extreme amounts of time which is inefficient and less effective.  In fact, ignoring the principle of first implementing simple and quick strategies will make solving harder problems even more time consuming.  This is because harder problems usually involve several sub-problems, sub-sub-problems, and so on.  And if for each of those sub-problems, you initially implement a sophisticated and time intensive strategy, then solving your original problem may be so time intensive that you may never successfully solve your original problem.

Being un-thorough during problem solving can be problematic because it can be hard to remember what you did and didn’t fully do.  Keep a careful record of strategies you avoided implementing due to being un-thorough in order to consider implementing them in future if your current approaches fail to solve the problem.  You will eventually need to be thorough in the “thorough problem-solving method” (after the notes) if you don’t find a solution in the faster less thorough methods.

†† A “neuron marker” is a neuron marker in your brain that receives information you discovered while solving your problem.  If the information would make the plan that you abandoned and marked with a “neuron marker” seem very promising, then the “neuron marker” would tell you to re-implement that old strategy.  Doing this is being opportunistic which is important for problem solving.  This is a sophisticated way of saying that you should put the idea in the back of your mind and take advantage of the idea if opportunity presents itself.  You should save (create a neuron marker for) any sub-problems you solved that didn’t currently help a certain plan, but may somehow be a useful “resource” later towards solving your problem.

‡ 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.

‡‡ The reason that you wait until now to break down entities in the problem is because you don’t want to break parts of the problem down unless it's necessary to take such desperate measures.

††† Starting over means that this method may take more time to implement.  This is one reason the method is set up to be somewhat fast.

Other Notes

The reason why you do the resource search and relate routines to some degree before forming a plan is because the information obtained from these routines helps you significantly when creating a plan.  You should do resource search and relate routines until you feel comfortable  trying to form a plan.  You may have to return later to search for resources and do relate routines during the problem-solving process to help you follow your plan, alter your plan, create other sub-plans, or create another plan if you have difficulty creating a successful plan.  Then after forming a plan, you can return to resource search and relate methods as you see fit for a need.

Also, when searching for information (resources) that might help in the problem-solving process, do not search excessively for information if you feel that you have already found enough information to construct promising plans for solving the problem.  You can return to searching for resources later if doing so becomes necessary.

You may have to break down parts of the problem that came from other parts found by breaking down even other parts earlier.  An example of breaking what you want into its parts given by Polya [7] was if you want to find a circle you need to first find its center and radius.  Later when you discover more known information, you may try to put what you want into a form that allows you to use this newly available known information.