Mathematical nature of the algorithm.

Of course, the most cloudless part of my childhood is the kindergarten period.
Our kindergarten number 3 was located on the outskirts of the Moldavian town Calarash, in the heart of the Moldavian Codry.

A picturesque green landscape, a huge territory of a kindergarten, with fruit trees: apricot, peach, apple, cherry, plum, walnut.
The last year in the kindergarten before school, sunny summer days and the kindergarten teacher keeps us in the playroom, for an hour, every day, and makes us learn the alphabet, and count sticks, and also write how many sticks on the board in numbers.
What is boredom! Children of the younger groups are frolicking outside the windows, and we children of the preschool group, look at them with envy.
Well, here is the first class of the school, the continuation of an incomprehensible drill. What all this is for – reading and writing letters, a combination of letters, syllables, words, making sentences, counting sticks. It’s like starting an engine that resists and drains the battery. Each person is like a separate motor, but the battery is consumed differently for each motor. A teacher in primary grades who reads all subjects is an accumulator running out of efforts to invest in a particular student. Very quickly, he has students-leaders and lagging students. This is natural.
Gradually, I got involved in the educational process, including doing homework and accepting reality. Although, I still did not understand, like many of my peers, what is this for? It is boredom when you have to sit quietly for 45 minutes in class, listen to the teacher, do what he says, and look forward to the school recess bell.
Here, then, I learned to count seconds in my mind, reaching 4-digit numbers when the lesson was not interesting to me, impatiently waiting for the school bell at the end of the lesson.
For me, the only subjects with meaning were the lessons of labor and natural history, which somehow even intertwined with each other – the flowering of flowers, fruitful trees, how rain clouds form, why does snowfall, making bird feeders, transplanting flower tubers.

At the end of the first grade, almost at the end of May, when the air is filled with the aroma of blooming gardens – the teacher already asks us to learn the multiplication table by heart. Oh, how I started to hate cramming then!
Okay, you can learn the poems by heart, the lyrics. But, how can you learn by heart – something that is not clear? How does it work?
Over the summer, I, all the same, mastered the multiplication table, thanks to the property of counting seconds in mind, countless times “from call to call of the school bell,” that had generated in the lessons of subjects that seemed boring to me. It seems now paradoxically, but among these lessons – there was also mathematics!

I mastered the multiplication table without memorizing it automatically. Still, by adding to itself the multiplied number in my mind by a certain number of times equal to the multiplier of the product – this was my first algorithm, with which I was able to master the multiplication. I have consciously come to this, and not by the method of automatic cramming.
When, already in high school, I knew that this is exactly how the arithmetic part of the microprocessor works in a computer. When performing a multiplication operation, i.e., the product of 2 numbers, this is essentially multiple additions of the multiplied number a certain number of times equal to the multiplier.
The leap in my studies took place in the middle of the second grade, when, on my own free will, I began to read books, the study engine started up!
I began to solve mathematical problems according to the textbook independently. Perhaps this was facilitated by the fact that the textbook’s talented creators put meaning from real life into the conditions of the mathematical problems, unlike the tedious technique of counting sticks as an introduction to mathematics. By this fact, the meaning of the mathematical problem was no longer for the school child abstract.
I was fascinated by solving problems in the textbook, and I independently mastered the material in mathematics, outstripping my class for several months ahead.
The speed of thought exceeding its presentation in a conversational form – led to a misunderstanding of me by those around me. The teacher made me read and talk slower and at the same time, unfortunately retraining from left-handed to right-handed – led me to stutter, this later, it was scientifically proven that in no case should you do this!
The turning point in my studies occurred in the 3rd grade. I became a leader in mathematics, displacing excellent students.
In the 4th grade – I became the undisputed leader in solving mathematical problems. There I first got acquainted with the term “Algorithm for solving problems.”
In some way, I solved the problems with my algorithms, different from the standard algorithms for solving problems given in textbooks, for each problem – I had my algorithm for solving problems. This approach to solving problems was seen and appreciated by the high school mathematics teacher – Irina Vasilievna Mosheiko.
It appeared in my mind, the founder of algorithms, Al Khorezmi – he was repeatedly mentioned by this teacher along with Archimedes, Pythagoras, Lobachevsky.
Al Khorezmi was better known as a systematizer of solving algebraic problems – according to a method that guarantees, after a certain number of steps, the solution to the problem.
For example, in the good old book for extracurricular reading on mathematics by I. Depman “Stories about new and old algebra,” there is such a verse characterizing two of the key rules of Al Khorezmi “Al Jabr” and Almukabala “:

Cancel minus terms and then
Restore to make your algebra;
Combine your homogeneous terms
And this called muqabala.

In my opinion, it was lucky to me with the teacher; she in every possible way approved of my non-standard approach to solving problems, including the search for an individual algorithm for solving the problem.
Once my father, returning from the parent meeting, told me that he had such a conversation with Irina Vasilievna (her classroom was near to our classroom): “You are the father of which school child?”
The father answered – “I’m Gennady’s father.”

Irina Vasilievna: “Great, I have long wanted to talk to you about your son’s mathematical abilities. I believe that he should continue his higher education at the university’s mathematics faculty after school graduation. At the moment, I strongly recommend your son to enroll in the school mathematics circle … I also recommend subscribing to the “Kvant” magazine for your son. ”

Once I brought to math circle – my formula for a problem, of an arithmetic progression. The algorithm for solving the problem was compressed into a formula (also essentially an algorithm).
The essence of the concept of a problem led me to a natural step – the solution of the problem, but the solution, i.e., the algorithm turned out to be quite long, and I decided to optimize it – reducing the number of steps to a minimum.
As a result, I got the formula:

d = (2 / k) (S (n + k) / (n + k) – S (n) / n)

The teacher asked me – where did this formula come from, which textbook I copied it from, in the school curriculum – this formula was not there, and for all the time of her teaching experience, she had not met such a formula.
After making sure that the formula works, the teacher instructed: publish the formula in the math circle’s wall newspaper. The next day, a wall newspaper was posted at the school – with capital letters – “Eureka! According to the initial data, a formula has been derived to find the step d of an arithmetic progression, according to the school’s math circle: the sum of n + k members and the sum of n members of the arithmetic progression!
This is one example of how the algorithm for solving the problem was found and optimized.
The wrong solution to the problem – is in the algorithm for solving the problem. But to create the correct algorithm for solving the problem, one must clearly understand the conditions of the stated task/problem.
Examples related to an incorrectly composed algorithm will be given in the following blogs.
Algorithmic thinking is the guarantee that the algorithm for solving the problem will be found.
It is worth emphasizing that many of my classmates in the class used my kindness in gratuitous help in solving mathematical problems.
What seemed easy to me – was very hard for others. Nevertheless, I always eagerly explained the solution to a specific problem and was happy with my classmates’ success. My part of help in mastering the educational material contributed to this.
Of course, some classmates automatically copied my solution to the problem, not bothering to understand how I came to the solution of the problem, i.e., creating the algorithm itself for solving the problem.
Sometimes it got to the point of absurdity when more than half of the class rolled the control tests written by 2-3 classmates. There were also purely mechanical errors that did not change the algorithm’s essence for solving the problem, but they became the reason for lowering the grades.
Now imagine how the class floor, having written off someone’s control work with errors, gets a solid three.

As paradoxical as it may sound – I did not strive to get A’s. I just solved problems, driven by the excitement in solving a non-standard problem, how more difficult the problem was, and the more interesting it was.
Indeed, some classmates would be more suitable for the term “trempist” because they hoped to earn the highest mark on someone else’s hump, without investing anything in the knowledge of a specific subject (in particular mathematics).
For imagine, they succeeded in almost all subjects. Having mastered this quality from school and thus amassed a good certificate at the end of school, they continued to be “trempists” in life.
As an assignment in the comments, the reader can compose his own version of the “trempist” algorithm for getting good grades in all subjects.
The reader can also indicate – in what subjects, the trempist, not having good abilities and knowledge in them, with all his efforts, still will not be able to use the “tremp” and get a good mark?