Why are squares magical?

Magic (sometimes referred to as stage or street magic to distinguish it from paranormal or ritual magic) is one of the oldest performing arts in the world in which audiences are entertained by staged tricks or illusions of seemingly impossible or supernatural feats using natural means. These feats are called magic tricks, effects, or illusions.

Mathematical games have been used for a very long time to interest us in puzzles, logic and finding solutions. With the advent of computers there are many mathematical games on-line. Some people have become addicted to them in the hunt for the solution. Magic squares are an example of a mathematical game which has been known about for over 2 000 years and can be done with pencil and paper or on a computer or mobile device. A magic square is a grid of numbers where each row and column and diagonal adds up to the same number. Magic squares date back to at least 650 BC in China. They acquired a mythical significance and have appeared as symbols in works of art.

Legends dating from as early as 650 BC tell the story of the Lo Shu or the “scroll of the river Lo”.  According to the legend, there was at one time in ancient China a huge flood. While the great king Yu was trying to channel the water out to sea, a turtle emerged from it with a curious pattern on its shell: a 3×3 grid in which circular dots of numbers were arranged, and the sum of the numbers in each row, column and diagonal was the same: 15. According to the legend, people were able to use this pattern in a certain way to control the river and protect themselves from floods. The Lo Shu Square, as the magic square on the turtle shell is called, is a magic square of order three. This means a square of three rows and three columns.  Using the numbers one to nine only once, the solution is when the numbers for each row and column add up to 15 as well as adding up to 15 diagonally. There is in fact only one solution and it can be found relatively easily by trial and error. There are in fact 8 variants but they are symmetrical uses of the same number pattern. In our TriHex app you can try this out for free with the app totalling the numbers for you as you enter them.

Islamic mathematicians also knew of magic squares. The first magic squares of orders 5 and 6 are in an encyclopaedia from Baghdad around 983 AD. A magic square of order 5 has five rows and five columns and 6 has six rows and six columns. Simpler magic squares were known by Arabic mathematicians earlier than this and were used with magic letters to assist Arab illusionists and magicians.

Magic squares are associated with mystery and mythology and have also been found in European Christianity.

Albrecht Dürer’s magic square

Albrecht Dürer was a German painter and engraver of the late 15th and early 16th centuries. His works were renowned throughout Europe and were mainly religious. In one of his engravings Melancholia (1514) there is a magic square. It is an order 4 square with four columns and four rows.

Detail of Melencolia


This order-4 magic square is believed to be the first seen in European art. The sum 34 can be found in the rows, columns, diagonals, each of the quadrants and the corner squares (of the 4×4 as well as the four contained 3×3 grids). The sum can also be found in the four outer numbers clockwise from the corners (3+8+14+9). The two numbers in the middle of the bottom row give the date of the engraving 1514. The numbers 1 and 4 at either side of the date correspond to the letter “A” and “D”, which are the initials of the artist

Sagrada Família magic square

The Basilica of the Holy Family is a large Roman Catholic Church in Barcelona,  designed by Catalan Spanish architect Antoni Gaudí (1852–1926). Gaudí’s work on the building is part of a UNESCO World Heritage Site. Construction of Sagrada Família commenced in 1882 by architect Francisco Paula de Villar with Gaudí becoming involved in 1883.

Gaudí devoted his last years to the project, and at the time of his death at age 73 in 1926, less than a quarter of the project was complete. It is still not finished today. The Basilica has three facades. One of these and the first to be completed is the façade of the birth of Christ. Leif and I visited Barcelona and the Sagrada Familia in 2007 and found the basilica awe inspiring and still unfinished.  This is one of our photos of the façade of the birth of Christ.The interior of the church was amazing with columns and ceilings taking on the natural look of trees growingplant

The second façade looked much more modern than the first. It is the façade of the Passion or Suffering and although designed by Gaudi was completed long after his death.

In 1988, the sculptor Josep Maria Subirachs was delegated with the task of creating the sculptures for this facade. He worked on the basis of Gaudí’s preserved drawings. This began with the crucifixion in 1989. At a scene with three levels of images the sculptor shows the story of the Suffering Way from the last Supper to the death of Jesus. Subirachs gave a riddle to the viewer to solve, a magic square, its sum always is 33 – the age of Jesus at his death. Structurally, it is very similar to the Melancholia magic square, but it has had the numbers in four of the cells reduced by 1.

When I visited the basilica I knew nothing of magic squares and did not notice it. The third façade is under construction. This is the façade of Salvation. It is hoped to be completed by 2026 one hundred years after Gaudi died after being knocked down by a tram.


The four x four Magic Square has a fundamental set of 220 squares which means 220 solutions are possible. Four different squares can be produced from each fundamental square by switching rows and columns.




On our app “TriHex” you can try to solve the 4×4 magic square with or without hints. Give it a go! The app is called TriHex but all that is talked about in this blog are magic squares. Magic Hexagons are also there but that is a story for another blog.




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