A lesson from an 8-year old
I’m going to talk bit about solving problems.
Two of my areas of speciality are software development as well as mathematics but I find problem solving techinques in math and software are also applicable to more mundane day-to-day problems in interpersonal relationships, politics, ethics or any other category you could chose.
Being a card-carrying nerd, I’ll start with math and software. In modeling or solving real-world problems mathematics often intersects with multiple domains, including software.
A few more complex examples on the top of my mind include:
- modeling or predicting climate change.
- the rate of growth and spread of airborne viruses.
- comparing investment scenarios in order to chose the better of multiple financial decisions.
I’ll cover each of these in this series but first, I’ll with a much simpler example of problem solving and basic principles.
Schools throughout the world, but particularly in the United States, tend to favor the memorization of facts and algorithms for solving specific types of problems. Many adults may still remember algorithms and formulas they learned as a pre-teen; for example, the Pythagorean Theorem. For a right triangle, where one angle is 90°, the square of the side opposite the 90°-called the hypoteneuse= is equal to the sum of the squares of the other 2 sides. c²=a²+b².
Most adults won’t remember why this is true or how to prove that it is true; something, young people were expected to be able to do more than 2,500 years ago in some cultures.
Jame Garfield, as an Ohio congressman and before his election as a US president, contributed an original proof of to the hundreds listed in “The Pythagorean Proposition”.
His proof was unique and one of a graphical nature. Of the almost 400 recorded proofs, many are of this type; especially some of the earliest discovered by the ancient Greeks.
Which brings me to the first principle of problem solving:
There are often many ways of solving a problem; all equally valid. Avoid the pitfall of looking for the “one true way”.
Engineers are discouraged from constantly re-inventing the wheel, or making expensive changes that offer little return of investment. At the same time, engineers are encouraged to always be observant of changes that will make things easier, cheaper and/or faster to create.
When teaching programming skills to beginners, sorting things is something we cover the first few days. Many thousands of brilliant programmers constantly research the problem and every few years, someone will publish a new more efficient algorithm. The older algorithms have names like: Shell Sort, Bubble Sort, Quick Sort, Merge Sort, Heap Sort,. Sorting is so important and vital, it’s an example of a thing so mundane is worth occasional re-examination.
Sometimes great advancements are made by youth and beginners simply looking for an easier way to solve a tedious problem.
Case in point: A few hundred years ago, there was an 8 year-old boy named Johan Carl Friedrich Gauss (I’ll just call him Fred). About 1785 (plus or minus a few years) his teacher gave the class the assignment of adding all the integers between 1 to 100: 1+2+3+4+5+ ……. +98 +99 + 100.
The “true or correct way”, I assume, was to add them sequentially, carrying the sum forward:
Fred, on the other hand, just blurted out “5,050” as the answer. I assume the other kids were just past the point of adding 1,2,3 and reaching the intermediate sum of 6 when the smart-ass poor kid blurted out his answer Some versions of the story indicate the professor was irritated and assumed Fred had somehow cheated. When the professor asked Fred how he had so quickly solved the problem, the boy essentially said that he had visualized all the numbers between 1 to 100 and the same numbers, 100 to 1 arranged side by side and that the pairs were all 101.
1 + 100 = 101
2 + 100 = 101
3 + 100 = 101
98+ 100 = 101
99 +100 = 101
100+ 1 = 101
The total of all pairs is obviously 10,100 since there 100 pairs, each 101
All integers 1 to 100 is a set. Integers 100 to 1 are a set. These 2 sets have exactly the same elements, but in different order, therefore all the integers added up in either set is half the the sum of both sets, which we already determined is 10,100. Thus the answer is 5,050. Incidentally, Gauss grew up to make major contributions to number and set theory. For decades, educators have attempted to instill in K-12 kids the fundamentals of Fred’s work, known in the 60’s and 70’s as”The New Math”, and more recently as “Common Core Math”.
Children in K-12 are often asked to simply memorize, without questioning and demonstrating what they’ve memorized in quizzes and exams. Gauss, as an 8 year old visualized that the sum of any arithmetic sequence can be calculated by adding the first and last terms, multiplying by the number of items and dividing by two. Sum=numItems (a1+a2) / 2, a1=first item, a2=last item. Now you know the source of this formula.
Kids and adults with some degree of autism, often find solving certain types of problems in the canonical or traditional ways difficult, boring or challenging. They will often discover alternative ways that work better for themselves. My own experience has shown me that all humans and many animal species have the same marvelous adaptive capability to solve problems. In this series, I’ll start with math and software related problem solving but I’ve found the same principles work in many other domains.
A 4-year old child who’s told, “no cookies till after dinner” or “no wild animals in the house”
, will often find a novel method of solving their problem the parents never imagined. The 16 year old daughter who’d told “no” to dating an 19-year old may also invent a problem solving scheme that would make Ethan Hunt of the Mission Impossible franchise envious.
In the next 2 posts, I’ll explore the principles of decomposition and pattern discovery, both of which are part of the same Gauss story we’ve already discussed.