Ten ways a Problem is Difficult

What Makes a Problem Difficult?
by
G. Donald Allen

Introduction. All of us solve problems daily. At home or office, from balancing the corporate books to teaching to researching theoretical physics, and even managing family affairs, successful days are those with problems well solved. Many problems are routine or even easy, but too many arrive with a measure of difficulty. So, what’s difficult?

Let’s begin by indicating that “difficult,” when applied to problem-solving, is difficult to describe. We explore several possibilities, for those of you already good at problem-solving. That is, you know the general problem-solving steps (Appendix A) and have used them to succeed previously. Certainly, experience is extremely important. Lack of experience can make almost any problem difficult as we’ve learned from history – our own included. Also, and of great importance is the ability to recognize a solution if by chance you stumble upon it. Organizations of all kinds need a “fixer,” an expert at solving problems. Here we are in the real world, well beyond ninth-grade math class.

Interestingly the notion of “cracking a problem,”  is applied especially to solving a difficult problem, metaphorically compared to cracking a nut.

Difficulty. Everyone everywhere needs to solve problems. People have problems, as do groups, cities, corporations, and governments. This makes the term "difficult" even more difficult to consider, much less define. We’ve identified ten categorical possibilities for what makes a problem difficult. In particular, Items 1-5 are relevant to the schools, science, and all of us generally, while items 7-9 consider humanistic factors like emotion. Items 6 and 10 are rather special for the advanced problem-solver.  Note, only a few are of the math or science variety yet are included in the general schemata.

1. Difficult as in others can’t solve it. Sometimes called an open problem, it requires you need to be armed with all the literature, expertise in the problem area, strong confidence, and most importantly good ideas. As well, you need a period (even a moment) of inspiration, but from where it comes can vary wildly. Such problems are certainly outside-the-box problems.
   
   
2. Difficult as in the problem may be unknowable. These are hard to identify. After all, how can you know what is unknowable? Perhaps solving the problem of the universe's origins is among them. Sure, there are theories, but there is no certainty. Another big one is “What is truth?” For millennia, this problem has had so many solutions, it has no solution.
   
   
3. Difficult as in too difficult for you. This type of problem takes you right smack up against reality. Even expert problem-solvers need to learn to understand when some problems are beyond their reach. Good advice is that perhaps it is too difficult if you haven’t solved it after two years of steady work.
   
   
4. Difficult as in not your cup of tea. Even the great Einstein could not solve simple political problems with the same ease he created relativity. As for the rest of us, many problems are simply not within our capacity to solve much less understand. The best problem-solvers know what kinds of problems are simply beyond their scope. Included here are also those problems totally new to you. Without experience or previous knowledge, the problem automatically qualifies as difficult, if only because you may not be equipped to consider all its ramifications.
   
   
5. Difficult as in you don’t have a large enough “toolkit” to crack it. The best approach in this case is a long and intensive study of the problem and the problems surrounding it. This is how Andrew Wiles cracked Fermat’s last theorem. Took seven years. On the other hand, many simply are not willing to walk the walk, continuing to try old methods that haven’t worked and continue not to work.
   
   
6. Difficult as in “wicked.” (Appendix B) Such problems, usually with an attached optimality and vague condition, are so complex with so many variables that different answers pop up as you finagle the conditions, parameters, and models. These problems are rampant in every complex society and large corporation, and the solutions adopted are rarely met with universal acclaim.
   
   
7. Difficult as in emotional. Here the solver is often tripped up by the very emotions needed to solve it. This can be confounding to even the most analytical thinkers among us.  Emotions love to conflict with each other. They conflict with other difficulties, as well, as contravening clear logical thinking. Emotion solutions are always present in popular music with titles such as “All You Need Is Love.”  Often incorrect solutions are determined again and again. Furthermore, solutions can cycle round and round at times coming close to a previously rejected solution. One could call these solutions generating human fractals. Even worse, the emotional solution may be precisely what’s needed, much to the dismay of those emotionally challenged
   
   
8. Difficult as in affected by social factors. These include all the political, religious, political, and cultural factors, basically summarized as “You believe this and I believe that” within groups. As with personal factors, these could be combined with emotional factors, but they occur in group form. Best known to all new watchers, particularly political news, they seem to have accepted them and hope to vote for “group” preferences. Worse, they become problems leading to governmental transformations. Definitely related to human belief systems, social factors create a large block of what appear to be problems of society or humanity, in general. These would be considered by many as the largest group of difficult problems.  Clearly, they cause multiple conflicts between and even among religions. Nowadays, problems of religion, including taboos, pale in comparison to other social and political problems.
   
   For example, if all climate activism was accepted by conservatives first, climate activists today would substantially be more conservative, rather than liberal as most have become. Of late, some of these difficulties are linked with individual rights of personal action projected upon the wide social screen. On a larger scale, such problems connect with governance on the whole. Figure into this group are solutions offered because of mixed methods or mixed groups. For example, when science and politics suggest different solutions, this compounds difficulties.
   
   
9. Difficult as in affected by personal factors.  You are under stress, angry, stubborn (like me), bitter, constrained, exhausted, prickly, picky, sensitive, pedantic, rigid, or dozens of other states of mind. These truly affect the level of difficulty for problems at hand together with resultant solutions. These encompass emotional issues but dive deeper toward proclivities such as time availability, a sense of security, and the normal in situ restrictions. Summarized these through the contemporary philosophy of postpositivism, which asserts that in many circumstances the person is involved in problem-solving and thus the solution. Therefore, for certain types of problems, we suggest, with trepidation, that objectivity is an illusion. Even artificial intelligence (AI), regarded as the ultimate escape from emotion is affected, by dint of its human programming.
   
   
10. Difficult as in undecidable. This type of problem was identified only a century ago by logician Kurt Gödel. He showed that in a sufficiently rich system, there are propositions that can never be proved either true or false. This discovery was to the dismay of mathematicians and in other subjects.  This takes the unknowable to the next step of being undecidable. One could argue that undecidable is a special form of unknowable, can occasionally it can be proved. One humanistic consequence of this sort of problem brings up the notion of compromise as a substitute for an undecidable situation. Problems of religion are also undecidable, though not in a strictly rational or universal sense. Such problems combine the unknowable, emotional, personal, and even informational.

Further Difficulty Factors. We have excluded problems solved but with unintended consequences. These may or may not be difficult, but rather “solution” solves only one part of a problem but creates other problems. The difficulty here is that the solutions are simply not thought through. Indeed, some of these errant solutions may themselves generate truly difficult problems.

Another type of problem but not on the list is that of random or rapid decision problems. Some issue/problem occurs. You must respond quickly without even a moment of thought. You apply randomly, instinctively, or even intuitively. While we all react, this sort of problem “difficult” cannot be included in our top ten because no methodology, rationality, or consideration can be applied. Solving on the spot is paramount. Only later comes knowledge of an evaluation of good or bad. Experience, processing speed, and general intelligence factor in. However, the problem may or may not be difficult, as there is no time for analysis.

A final type, “Difficult as in vague,” is not included in the main list because these are problems of statement, and this leads to difficulties of interpretation more than the difficulty of solutions. The term is anathema to well-posed problems, as they are unclear and suggest countless authentic solutions. For example, consider the vagueness of language, which of course is a vague instrument of communication. For example, if tasked to find a good solution to solve the problem of drinking and driving, the problem-solver might come up with many ideas from jail to counseling to big fines to any of many other possibilities. The word “good” is the culprit here. It corresponds to other terms in which some optimality is required. Wicked problems almost always have vague components.

Conclusions. One certain takeaway from all this is for us to consider whether we have the tools to solve the problem. An old maxim is, “The poor craftsman is the first to blame the tools for a poor job.” Yet, tools are but an extension of the craftsman, in the present case knowledge and the ability for use by the problem-solver. Few problems are solved with poor tools, relying on some clever insight from nowhere. Putting this in the vernacular of the current language, one cannot think outside the box until they have well learned to think inside the box. Problem difficulty is only part of this. Some problems have their own traps tricking one toward solutions that miss the mark, just as the hammer missing the nail.

The CEO or any leader tasking their staff to solve relevant and important problems should take note if their staff has the right tools, and how inappropriate or wrong solutions can be offered. The parent or spouse should consider whether they are swept away by emotion in framing solutions. The priest should consider carefully whether the particular scripture truly does apply. And all should know the tools to help know and then crack difficult problems.

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Appendix A. Normal problem-solving steps.

There are countless methods to solve problems. All problems, not just math problems, are covered. One particular five-step list follows.

A.     Identify the Problem. It is critical to have a clear statement. Trying to solve a vaguely stated problem is doomed to failure. 

B.     Consider the tools of data/knowledge/literature/tests/logic/algorithms available to solve the problem.

C.     Using these tools and your experience, generate potential solutions.

D.     Choose the best solution – in the case of multiple solutions.

E.     Implement the solution you've chosen.

These sound both simple and routine, but each step leads to its own set of problems, and each of these may involve internally the same steps. Yes, steps within steps. For example, even clarifying the problem can be a problem itself. Many more detailed problem-solving lists are available online.

Appendix B. Wicked problems.

Wicked problems have always been with us. Only recently have they been defined, clarified, and exemplified. The word "nasty” is certainly appropriate, because what else can you say about essentially unsolvable problems that must be solved? Some family problems, particularly those involving teenagers, are clearly wicked. Here are a few of the general criteria. If you want a particular real-life problem to consider as you browse the list, think about building a beltway around a city from scratch.

a.     There is no definitive formulation of a wicked problem

b.     Wicked problems have no stopping rule. That is, you can keep solving. Eventually, an action plan must be constructed. Only then does the solving stop.

c.     Criteria for solutions to wicked problems are not true or false, but good or bad.

d.     There is no immediate and no rigorous test of a solution to a wicked problem.

e.     Every solution to a wicked problem is a "one-shot operation." There is no opportunity to learn by trial and error.

f.      Wicked problems do not have an exhaustively describable set of potential solutions, nor is there a well-described set of permissible operations that may be incorporated into the plan.

g.     Wicked problems are generally unique.

h.     Every wicked problem can be also considered a symptom of another problem.

i.      The existence of a discrepancy representing a wicked problem can be explained in numerous ways. The choice of explanations determines the nature of the problem's resolution.

Appendix C. Consider this short list of current difficult problems, most of which have no true solutions.  Of course, there are theories, beliefs, and positions on most of these,  but all without consensus and few with viable solutions as most have been with us for years if not centuries.

Survival

2.     Starvation

3.     World peace

4.     War

5.     Misinformation

6.     Environment

7.     Food supply

8.     Poverty

9.     Fair Trade

10.  Reduction of Debt

11.  Unemployment

12.  Economic Disparity

13.  Economic Policies

14.  Human Relationships

15.  Communication

16.  Decreasing Conflict

17.  Disease and Human Suffering

18.  Education

19.  Artificial Intelligence

20.  Cryptocurrency

21.  Cryptography

22.  Government Surveillance

23.  Technology

24.  Energy

25.  Refugee Crisis

26.  Resource Depletion

27.  Human Rights

28.  Water Contamination

29.  Population Growth

30.  Nuclear families

31.  Privacy

32.  Pandemics

We do see solutions offered regularly, often in a postpositive amalgamation of politics, science, and various other beliefs. Indeed, many would have you believe that politics must be included. Nothing could be further from the truth. Mixing two problem-solving methods (e.g. politics and science) more often leads to further difficulties.

 CEOs and others in business have their independent set of problems, many of the difficult variety. They vary from supply chain, transportation, dividends, cost points, advertising, assembly line, import/export, personnel, and dozens more. Make a list. You may be horrified.

We, as in us folks, also have a passel of personal problems. Many, if not most, involve human relations, money, and health.  Given below is a personal list of difficult problems I’ve encountered over the many years. If you have fewer than half of these, you’ve been lucky in life. There are lots of little problems. On the other hand, you may have your own list of big ones.  Perhaps a bit cynically, our politicians have entirely similar personal problems yet are challenged to solve those on the lists above.  And we know politicians just love talking points.

  Children

2.     Rebellious teenagers

3.     Cancer

4.     Disease, e.g. heart disease

5.     Job satisfaction

6.     Job search

7.     Job skills

8.     Making ends meet

9.     Love

10.  Hate

11.  Debt

12.  Household repairs

13.  Insurance

14.  Dating

15.  Depression

16.  Friendship

17.  Sleep

18.  Estranged relatives

19.  Secrets

20.  Forgiveness

21.  Death

22.  Quality of life