8 June 2018

Magic square

A magic square of order \(n\) is a square table, made up of cells, with the same number of rows and columns. In each cell is written a natural number, so that the first \(n^2\) natural numbers are written and the sum of the numbers in each row, each column or each of the two diagonals are equal. At this value we call the magic constant. The magic constant of a magic square depends only on \(n\). Thus, for a magic square of order \(n\), the magic constant is \(\frac{n(n^2+1)}{2}\).

Let us consider a magic square of order \(3\):


Note that the sum of each row, each column and each of the two diagonals is equal to \(15\).

Magic squares are special, but why are they called magic? It seems that in ancient times they were connected with the supernatural or the magical world. The first magic square record was discovered in China, dating from about \(2200\) BC and was called "Lo-Shu". There is a legend that says that the Emperor Yu saw a magic square on the back of a turtle in the Yellow River. The black nodes always represent even numbers, while the white nodes always represent odd numbers.


The first reference in the West about magic squares was found in the work of Theon of Smyrna, a Greek philosopher and mathematician. They were also used by Arab astronomers in the th to build horoscopes. The work of the Greek mathematician Moschopoulos in \(1300\) helped spread the knowledge about magic squares.

Magic square on the Sagrada Familia church, Barcelona, Spain.

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