An Introduction to Methods and Heuristics for Problem Solving
Most methods and heuristics for problem-solving skills can be summarized under these categories:
These may seem simplistic, but there are a lot of details about how to best go about using these skills in an efficient and effective way. This section will discuss in great detail how to use these skills. The first skill in the list above will be referred to as Know-Want-Relate, and the second skill in the list will be referred to as Creating a Plan. Usually when solving problems you want to start with skill #1, and then move on to skill #2 when you feel you know enough about your problem to do so.
Skill #1 Know-Want-Relate
Consider what you know, focus on what you want, and then later create a plan (in skill #2) to consider how to relate what you know to achieve what you want.
Do not only think about what you want; concentrate on it. This is the goal of everything you are doing.
There are many branches of mathematics, and each branch has an enormous amount of theorems, definitions, and other material. Much of this material may be needed to solve your problem. If your problem is in a textbook, then usually the information in the textbook before the exercise should be sufficient, otherwise, you may need to investigate the resources available in the specific fields of mathematics that concern your problem. Insufficient resources can make solving your problem difficult unless you either find the proper resources, or discover the many needed resources on your own. The latter idea would be a lot of unnecessary work when you could first study the subject first. Of course as mentioned earlier, be sure that the resources you use are correct.
Other important considerations involving this strategy:
-Any time you discover truths you know about the problem or that may relate to your problem, write them down in a list of “known information”.
-Consider the implication of each part of the given information in the problem in every possible combination with other parts of the given information in the problem. Do this using defeasible reasoning.
-Keep in mind the conditions and restrictions to the problem.
-Consider if and how the problem might be reconstructed under a different type of model. (see method 11 in catalog) For example, a difficult algebra problem may be easily solved under the framework of geometry. Or a differential equations problem may be easily solved under the framework of abstract algebra and differential geometry. Also many problems can be restated in terms of graph theory where there are rich useful theorems that can be applied.
-Consider searching for a useful theorem that may help to solve the problem or a part of the problem. Theorems that relate to entities of your problem, its conditions, and its restrictions may be the best theorems to consider first. Maybe some of the theorems you find don’t immediately help you solve your problem, but might help in a part of your problem later. Consider seemingly true variations of the theorems you want to apply and see what difference in the stated theorem would be useful for your problem. If you find a useful variation of a theorem that seems true, then try to prove the new theorem in order to use it.
-Parts of the problems that are hard to solve by directly using other theorems will likely have to be solved by breaking down pieces of given information into their defining parts and solve the problem using other theorems or strategies that apply to these defining parts. Try to recombine these defining elements in some new manner. Maybe recombine them into a new sub-problem that helps to solve the original problem (by working backwards).
-Try to find truths implied from the given information that might be helpful in solving the problem. Add to your list of “known information” those truths you find from that might help you achieve what you want to find. After finding new truths, look at the new list of “known truths”, and re-analyze how elements in combination from your new list of “known information” could help you get what you want to achieve.
-Look for a pattern. This method is quite involved. (See method 12 in catalog)
Experiment to search for possible truths and relationships that relate to your problem:
(see method 7 in catalog)
a) Find the right questions to ask about what statements concerning your problem are true in order to find the needed truths through experimentation and searching.
When you know many truths and relationships that pertain to your problem, you have more power to manipulate and control the mathematics of your problem. But you cannot begin to know these truths until you ask the right questions. The right questions are those that pertain to information that would likely relate to your problem, and whose answer would likely give you vision on how to solve your problem or provide possibly needed information.
b) Search for truths that may help you solve your problem or answer important questions formulated in (a) by experimentation with mathematical entities such as numbers and by using other thought experiments.
Often when solving problems, there are other mathematical statements that if true would help you to solve a part of your problem (This can guide some questions in part (a)). Often you need to search for truths that relate to your problem that could help you find a solution (lemmas in the case of a proof). You may need to experiment with different numbers or relationships between mathematical entities.
You want to consider experimenting with extreme types of relationships to make sure a general relationship has a high probability of being true in many given extreme circumstances.
Test to see if any suspected truths you hypothesize contradict each other. Try to find counter-examples that could contradict your hypotheses. If you do not find a contradiction after serious thought, then you may try and prove the statement you need using problem-solving skills. Then you may find it is difficult to prove your hypothesis, and search for a change in the condition that may be necessary in order to prove your hypothesis. Try to find such conditions that will still help you with your original problem, but allow you to prove a variation of your hypothesis. This process could be demanding because the solution to your sub-problem is needed by your original problem. But solving your sub-problem may have its own sub-problems that go on and on. For example, Andrew Wiles’ proof of Fermat’s Last Theorem was over 100 pages because of so many problems needing sub-problems solved.
c) Continue searching for needed truths (resources) until you feel you have sufficient information to solve your problem. Once you have sufficient information to solve your problem, it is a waste of time to continue searching for truths that may relate to your problem.
Andrew Wiles compared proving Fermat’s Last Theorem to moving around in a dark mansion with no light [49]. When beginning to solve problems, you start as if you are blind searching for truths and relationships in a room. You have to investigate by feeling around to find out what truths and relationships exist that could be useful to you. Similarly when solving a problem, you need to know what mathematical truths exist that may relate to your problem.
One of the most common methods to relate what you know to what you want is to Work Backwards. (see sub-method 2d in catalog)
Working backwards often decreases the number of solution paths to solving the problem that you have to search through. Often when people try to solve a problem, they will randomly guess solutions which may never come close to a true solution, and may require in-numerable possibilities to find a solution.
For example, suppose the problem you want to solve is that of obtaining the number 24 using all four numbers: 5,5,5,1 with the restriction of using only the four basic mathematical operations of multiply, divide, add, and subtract with any order of desired operation. There is a large number of ways to perform such operations and you may work for a very long time trying to find 24. But suppose we assume we have already found 24. Then by working backwards from the last step, we then have the following second to last steps: 24 = 5x, 24 = x/5, 24 = 5/x, 24 = x - 5, 24 = 5 - x, 24 = x + 5, 24 = 1 - x, 24 = x - 1, or 24 = x + 1, etc. where x is found using the remaining three numbers. Solving for x, we now need to find which of those values of x is most plausible from the remaining three numbers.
Working backwards usually works well when you don’t have too many possible branches of possible sub-problems in the backwards direction.
Skill #2 Create and Follow a Plan (see method 24 in catalog)
When solving problems, you should search for the main steps that need to be taken to solve the problem. For example, if you are constructing a bridge, your main steps may be to design the bridge properly, obtain proper materials, and then implement construction. However, each of these main steps needs to be solved using more specific sub-steps. In other words, you try to break down the problem into solving a sequence of main steps that if you could solve, would give you the solution to the problem. This is creating a plan. The main plan seeks to define the main steps to solving the problem, where you will fill in all of the minor steps involved in solving the main steps of the problem later. Before working on solving the steps of a main plan, consider if each of the main steps seem promising to really have a method of solution. After creating a plan that seems promising, you should try to solve each of the main sub-steps of that plan to solve the problem. For each of the sub-problems in the plan, you will have to create a plan for each of those using all problem-solving skills to solve those and so on.
Try to create more than one plan if possible in order to choose the plan that seems the most promising. Consider trying to combine parts of other plans to make a less promising plan better than all current ideas you have for plans.
Look at the plan of other problems that involve similar mathematical relationships for ideas that may help you construct a more promising plan to solve your problem.
When searching for possible plans, maintain a list of all of your ideas, organize them and don’t throw any of them out. Some of your ideas could come in useful later if you have trouble solving your problem. Also, a part of another idea could become useful in combination with another one of your ideas.
-Be grateful for lesser ideas or hazy ideas and list them [52].
-Welcome unusual ideas as well onto your list. Do not squelch an idea just because it seems peculiar or unfamiliar to you. These ideas just might be how to solve the problem.
-Supplementary ideas can be useful to attempt the correction of a less fortunate idea [52].
-Consider any ideas that are incomplete [52] because any idea could lead you to another idea which could lead directly to the solution.
-Consider combining elements of various ideas to make a better idea using elements of both.
-Be sure to organize and categorize your ideas whenever you start to get cluttered with ideas and information.
Follow your inspiration with a grain of doubt [52] (Polya pg 184). However you should not doubt excessively. Do not squelch any ideas out unless you really know or can prove that a certain idea could never be possible.
Recognize the sub-processes of problem solving.
You may form a plan to solve your problem which requires solving various other sub-problems to achieve your plan. Your plan shows that the main problem has been broken down into for example solving problems (a), (b), and (c). But problem (a) may need the solution of several other constructed problems to find its solution. The same holds for (b) and (c). Recognize the sub-problems and their associated sub-sub problems to then try and solve those problems by starting over the problem-solving process on them. Choose the plan you create that seems to have the best combination of being the most promising to finding a solution and easiness to solve each of the sub-problems and associated sub-sub problems.
When executing your plan, pay attention to take advantage of any opportunities that would help solve your problem or simplify a more complicated part of your plan.
When carrying out a plan, you should be opportunistic if you notice a truth that can help you to either better implement your plan, improve your plan, or even to create a completely new plan that is more promising. You should take advantage of any opportunities you find to adjust or update the plan to make it more effective, efficient, and more promising to create a solution to the problem. For example, you may notice symmetry in equations you derive during the problem-solving process and notice that you may be able to take advantage of that symmetry to simplify your process or overcome obstacles to the problem. Always have a philosophy of opportunism when solving problems—meaning that if you find something that would likely help you overcome a difficult obstacle, then you should try to incorporate it into your plan.
When your plan seems to fail at solving the problem, or is difficult to implement after exhaustive work, or a serious flaw is found in your plan, then try to vary the problem and/or plan.
Vary the problem. (see methods 20 and 21 in catalog) If you can't solve a problem, then there is an easier problem you can solve: find it—George Polya. If you find your problem is difficult, then invent a similar problem that is more accessible. Solving similar problems will likely give much insight and information on solving your original problem. Here “similar” means that most details are the same, but some are different. Varying the problem also has the added benefit of removing the monotony in solving the problem and helps to maintain interest in the problem. Again, interest is key to solving problems to keep your mind working on the problem.
Vary your strategy and change your plan. (see method 28 in catalog) Maybe your initial strategy doesn’t solve the problem, but shows you what does not work. Failure is a step towards success. We learn a lot about our problem through failure. When a plan or strategy fails, you need to change the strategy because working on a plan you have found not to work is a waste of time. After you spend a lot of time concentrating on an idea, plan or point with few signs of progress, start to concentrate on something else. However, do not vary your strategy when you are progressing well, or have signs of progress.
Varying your plan also has the added benefit of helping you maintain interest in the problem to not be monotonous. As mentioned earlier, interest is key to helping you solve problems, because otherwise your motivation to keep working on the problem may dissipate.
Pay attention to when you have signs of progress towards solving the problem. If after much work you do not have good signs of progress, you should probably vary your strategy, or vary the problem.
Choose your problem-solving path according to the signs the problem-solving path you are taking gives you in order to avoid wasting time implementing faulty strategies. Sometimes the method you use to solve the problem gives you very few signs, so you must not ignore even the smallest of signs.
If you are experienced in knowing what is a good sign of progress, then you waste less time on unpromising paths. This is one reason why practicing solving problems helps you become better, because you learn to recognize signs of progress and waste less time on unpromising paths.
Sometimes there are misleading signs of progress, and you eventually hit a brick wall when solving your problem. Sometimes we are fooled by our inspirations. Trust your inspiration, but be aware, and pay attention to possible barriers ahead in the problem-solving process.
Some possible signs of progress to looks for:
-Statement A implies statement B, so statement B might imply statement A.
-Your method of attack has similarity or analogy to the correct solutions of other problems.
- Heuristic or experimental signs that are not certain, but signs nonetheless.
-Understanding the nature of the unknown in the paradigm of your plan. [52]
-If one or more pieces of data becomes connected to the unknown in some way, or if another clause of the condition is taken into account.
Distinguish between small steps and big steps [7] towards solving your problem in order to take into account the significance of each sign and step of progress.
When no other plans or ideas seem to work, then seek an original method of solution.
Original methods of solution that no-one has discovered or used in any other similar problems are harder to find. Knowing how to do this is what defines true genius. An example of an original method of solution is Cantor diaganolization.
Take some time off of the problem and sleep on it.
You may find solving the problem is easier after a night of resting on the idea and letting your subconscious do some work on the problem to organize the problem, concepts, and your ideas in your mind. Be sure to get quality and restful sleep.
Some Other Important Problem-Solving Skills
1. Create definitions for any of the following needs to simplify your problem-solving process: (see methods 3 in catalog)
-To give meaning to ideas and concepts that you have.
-To simplify clutter from commonly used sequences or combinations of ideas and concepts.
-To give a name to useful ideas or concepts that have problem-solving powers to prove theorems, or problem-solving powers to create mathematical methods and algorithms.
2. Use suitable notation, and create a diagram to assist your intuition of the problem. (see methods 4 and 5 in catalog)
Using complicated, cumbersome, confusing, or improper notation can significantly complicate the problem-solving process.
3. Find similar problems and their solutions. Consider if there is a useful similar method of solution to your problem. (see method 24 in catalog)
Also try to see what helped you in a similar situation. Here similar means that many details or aspects of a problem are the same as the similar problem.
4. Consider applying commonly used methods to solve problems that you learned during your analysis of problems solved in textbooks on the subject you have read. (see method 24 in catalog)
5. Think inside the box first, and when that doesn’t work, think outside the box. (see method 34 in catalog)
Thinking inside the box means that you restrict yourself to using your assumed preconceptions of the details of the problem or your preconceptions of methods of solution. Usually problems are simple, and using your preconceptions about your problem are correct assumptions you would expect to be true. However, sometimes your assumed preconceptions restrict you from finding a possible solution to the problem. Many restricting preconceptions of the problem can prevent you from using important information or methods to solve a problem.
6. Consider the use of auxiliary elements. (see method 13 in catalog)
Auxiliary elements are mathematical entities such as specified types of sets, functions, variables or other such mathematical entities that are created for the purpose of assisting in steps of a problem-solving process. Often solving a problem greatly depends on the clever creation of an auxiliary tool or created entity that assists in an important process of solving the problem.
7. Take advantage of any symmetry you recognize. Search for symmetry in your problem. (see method 16 in catalog)
Symmetry can be both algebraic and geometric. For example, x2 + xy + y2 is a symmetric algebraic expression in that x and y are interchangeable to give you the same expression. Symmetry is part of what you know that you should take advantage of to find what you want. You should also consider trying to put your problem into a form that has symmetry to exploit. More generally, you should put your problem into a form that allows you to exploit some kind of principle, theorem, other known information, or relationships.
8. Consider special cases of the problem. (see method 22 in catalog)
Many other important problem-solving skills exist, which are described in detail later in this book.