What Does the Way we Solve Math Problems Reveal About our Character?
Abstract:
In this article, through analyzing the thought process behind different solutions to a specific problem, I am going to make an attempt to delve into the inner psychological procedures that led to the respective different approaches and ways of presenting problem-solving ideas.
I deeply believe that the way each one of us chooses to tackle a problem reveals different character traits, neurological processes and learning choices. These distinguishable characteristics may also transpose themselves verbatim to the whole spectrum of life, and they are the underlying reason behind our life choices, our connections and our relationships.
I am going to present a problem proposed at an international competition. The solutions that will be introduced are the ones that Greek students (myself included) submitted to these contests. This was done on purpose, as it was easier to verify the results of the analysis conducted below. Although the data size is quite small, relevant studies tend to support my findings¹.
In order to comprehend the process below, a certain mathematical maturity is required — although attempt will be made to not delve into too many technicalities, and only highlight the main idea(s) behind each solution.
Problem:
Our first problem comes from the Balkan Mathematical Olympiad 2022². It was a problem 2, meaning that it is (comparatively) one of the easy problems of the contest. The Greek team submitted (at least) 2 distinct solutions to this problem, which we will attempt to analyze, along with an extra solution from the mathematical website AoPS.
First solution (submitted by Panagiotis Liampas):
Second solution (submitted by Orestis Lignos):
Third solution (published by AoPS user oVlad):
This problem belongs to the field of Number Theory, the field of mathematics devoted to the study of integer numbers. In this specific problem, we may instantly notice how technical and lengthy the three solutions are — to the extend that they probably all seem similar and indistinguishable from each other.
However, it is worth making the following remarks, that set apart each solution from the others:
Remark 1: Note that Solutions 1 and 3 both use a very specific and technical result. This result assists us on find the greatest power of a prime dividing a difference of two powers, and is the so-called Lifting the Exponent Lemma. This Lemma is quite prominent in Olympiad Number Theory problems — it is safe to say that the vast majority of the solutions that appeared during the actual contest made use of it at some point of the solution. However, Solution 2 does not seem to mention it, and this is because this solution actually proves this Lemma (see at the bottom lines of image 5) without never explicitly mentioning it!
Remark 2: Note that Solution 3 follows a different approach than the other two approaches. Indeed, this solution replaces 2023 with a squarefree integer k. For non-experts, the term squarefree means that such an integer is not divisible by the square of any prime. The reason that we picked k to be squarefree and not just an integer is evident from the solution path followed later on. Therefore, by replacing 2023 with a general number while maintaining the essential properties it carries, this solution achieves to tackle the problem in the fullest generality, while essentially using ideas similar to the other solutions.
Remark 3: Pay attention to the underlying differences between Solution 2 and Solutions 1 and 3. While the former solution adopts a brute-force approach — that is, a blind approach in which no care of tricky connections is developed — while the latter solutions approach the problem in a more coherent manner. Notice, for example, the repetitive nature of the Cases and Subcases in Solution 2 — these all fall under the same umbrella, but were instead tackled seperately, in contrast with the other two approaches.
My study asserts that these remarks are not pure coincidences. Instead, they all follow a similar pattern when it comes to how we tend to approach problems. Of course, this applies not only to mathematics problems, but also to the whole spectrum of life. Based on the above remarks, one may distinguish the following approaches:
Approach 1: The brute-force approach. I think this approach characterizes people that tend to be quite straightforward in tackling their problems or issues, and normally tend not to think much before pursuing a solution path. This approach is implemented in Solution 2 above — while a significantly shorter solution with essentially the same ideas was possible, I preferred to adopt a brute-force approach.
I even believe that this approach is also connected to tackling small tasks first instead of the whole problem. Another characteristic of Solution 2 above is that I tackle each case seperately, instead of tackling them all at once — and I believe that this way of thinking reflects a more general approach, underlined by a tendency to divide the initial, large problem into smaller, easier to handle subtasks.
Approach 2: The generalization. According to its definition, generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. In this context, generalization specifies the tendency to tackle a specific problem in a more abstract level, refraining from the more down-to-earth approaches and searching for more abstract connections, possibly reflected in other instances we have heard of or experienced, or even to our past hardships, failures and successes. This approach is further characterized by taking a step back when facing a problem, and viewing it in a more general and, possibly, objective manner.
This approach was implemented above in Solution 3. This solution is by far the most rigid one of the three presented above, and possibly also the shortest. An important reason for this is because the solver generalizes the initial problem, viewing the problem at a broader context, thus making the intial task just a small and specific subcase of this problem.
This approach has a lot of positive elements and reveals a lot of personality traits, too. Indeed, I believe that persons who tend to adopt such strategies are often more well-minded, with acuted critical thinking skills that allow them to make the insightful observations required to highlight the one specific property of the given problem that is prone to generalization (in our case, the fact that 2023 is squarefree).
Approach 3: This last approach matches with people that tend to meaningfully apply previous knowledge or experience to new, unsolved problems. In our case, this previous knowledge is, as we analyzed above, the Lifting the Exponent Lemma. This Lemma plays a vital role in developing the solution, but only Solutions 1 and 3 make use of it — on the other hand, solution 2 proves it from scratch.
I believe that this tendency reflects the approach of more experienced people, who are normally full of past experiences and have developed the connections and connotations necessary for bonding them with their current tasks at hand.
While relying on past knowledge characterizes experienced people, we must also mention that developing the solution from scratch reveals another unique trait — that of striving to make original breakthroughs and presenting innovative alternatives, which normally do not rely on anything said, written or developed before.
Conclusion:
This small investigation aimed to connect a mathematics problem and three different approaches to it with a more general approach to tackling life problems. Through specific examples, we developed the respective traits, habits and intentions associated with them, and connected them to a more general framework.
References:
[1]: https://www.verywellmind.com/what-is-problem-solving-2795485
