What is a magic square?

A magic square is a square array of numbers consisting of the distinct positive integers form 1 to n² arranged so that the sum of the n numbers in any horizontal, vertical or main diagonal line is always the same number, known as the magic number S = 0.5n (n² + 1). You can view Abiyev Balanced Square of any degree using the program below.

Input order of Balanced (Magic) Square:

Show numbers

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Being a physicist, why did I work with magic squares?

During my childhood years, I had written magic squares of 3rd and 4th orders by brute force. Considering the impossibility of writing the squares of higher orders by this method, and even the inexistence of such squares, I refrained from this entertainment. After about 50 years (1995), while I was working in Turkey, an article called “Magic Squares” from the “Mathematics World” (in Turkish) magazine caught my attention. It noted there that the magic squares are a mathematical problem. This piece of information overwhelmed my thoughts of childhood years, and I envied the fact that this problem will be solved by another mathematician very much. This mathematical problem consisted of finding a rule to easily write the magic squares of any order. On March 3, 1996, I solved this problem and I declared this algorithm in my book.

Abiyev's Balanced Squares

In 1995, after analyzing the information I gathered on magic squares from the main libraries of the world I came to conclusion that all magic squares are balanced. That is, given a homogeneous square board with point masses placed on the cells of corresponding numbers, center of mass of this system will be the same as its geometrical center.
But since my magic squares differ from all the squares written to date on certain properties, I named them Abiyev Balanced Squares. Note that, my algorithm allows to write magic squares of any order using any numbers (even symbols).

The importance of Balanced Squares

With their appearance and the patterns in the writing of the numbers, Abiyev Balanced Squares can't leave one unimpressed. These squares are the best example to the ingenious words said by Pythagoras regarding numbers and Paul Carus regarding magic squares:

“All things are numbers.” - Pythagoras
“There is no science that teaches the harmonies of nature more clearly than mathematics, and the magic squares are like a magic mirror which reflects the symmetry of the divine norm immanent in all things, in the immeasurable immensity of the cosmos and in the construction of the atom not less than in the mysterious depths of the human mind.” - Paul Carus Today, the problem is not the in writing of the magic squares, it is in finding their applications in different fields of science and technology. Using these squares in any mathematical model will likely produce new scientific achievements.