Who is the Prince of Mathematics?

One day, a maths teacher in a primary school wants to read and evaluate the exam papers of his students in the classroom. However, the students are making so much noise that the teacher cannot concentrate on what he has to do. Finally, he decides to ask his students a difficult problem that will keep them busy for a long time. His students already know how to add, subtract, multiply and divide. After thinking for a while, the teacher asks his students to add the numbers from 1 to 100 one after the other. Then, confident that this problem will be a challenge for them, the teacher resumes the evaluation of the exam papers. The necessary silence in the classroom has now been achieved. 

However, this situation does not last long and within a few minutes one of his students raises his hand and says that he has solved the problem and found the result. The teacher, in great surprise, asks the student what the result is. Because the teacher himself does not have the slightest idea what the result should be. When the student answers that the result is 5050, the teacher asks the student with even greater curiosity how he reached this result. The student gives the following answer to the astonished teacher;

‘First, in my imagination, I sorted all the numbers from 1 to 100 in a single line from small to large. I added the number 1, the number at the beginning, and the number 100, the number at the end. The result was 101. I looked at the remaining numbers on the line. There were numbers from 2 to 99 left. When I added the number 2 at the beginning and the number 99 at the end, I found the same number, namely 101. I thought that if I continued in this way, the sum of the numbers at the beginning and the numbers at the end would always be 101. Since I added the numbers two by two, in the end I would have 50 101s, half of 100. As a final operation, I multiplied 101 by 50 and got 5050.’

We can better understand how this student, who astonished his teacher, used his imagination and found the result with only a few operations by using the diagram below;

Since the total number of addition operations above is 100 / 2 = 50 and the result of each addition operation is 101, the sum of the numbers from 1 to 100 is 50 x101 = 5050. More precisely, we can use the following equation to find the sum of numbers from 1 to 100;

                                                             100 (1 + 100 )

1 + 2 + 3 + ..........+ 98 + 99 + 100 = __________________= 5050

                                                                       2

If you are wondering who this student is, let me tell you; the famous German mathematician Carl Friedrich Gauss. A question asked by a teacher to his students, thinking that they could not solve it in a short time, and the result of which he did not even know himself, caused a mathematical genius, the so-called ‘Prince of Mathematics’ Gauss, to make his first discovery while he was still a student in primary school.

This method invented by Gauss not only allows us to find the sum of numbers from 1 to 100 in a snap, but also to find the sum of numbers from 1 to 1,000,000 or even from 1 to 1,000,000,000 in a short time by performing only a few operations. 

To make a generalisation, no matter how large n is, to find the sum of all numbers from 1 to n, we need to add 1 and n (just as we add 1 and 100) and multiply by half of n (just as we multiply by half of 100). We can express this in a shorter form using mathematical symbols as follows;

                                                                       n (1 + n)

1 + 2 + 3 + ……….+ (n – 2) + (n – 1)  + n =  ______________          

                                                                                 2

This equation is known in mathematics as Gauss's formula and is just one of the formulas taught to us by heart in schools. However, its interesting origin story and the deep imagination and creativity behind its derivation, when well understood, gives a strange pleasure. As a result, there may be more than one way to solve a problem, and sometimes instead of exhausting and time-consuming methods, creativity and imagination can be used to find solutions in a short time with much less effort. The important thing is to think outside the box.