We have been fascinated with numerical patterns since we first gazed up at the stars. Our drive to find order in apparent randomness is probably why we have been to the moon, created complicated electronics and smashed atoms.  Even our interest in Sudoku is an expression of this same drive.

A 4x4 Latin square.

Magic squares are the original and more interesting version of sudoku that were invented over 2000 years ago. Unlike Sudoku, magic squares require you to have all columns, rows and main diagonals sum to the same value. This value is called the ‘magical constant’. You can only use numbers from one up to the number of squares in your puzzle and you cannot repeat a number. There are magic squares of any square size, except for 2x2.

An example of a magic square is on the top left of the main diagram. This magic square, used in a famous 1514 engraving by Albrecht Dürer (Photo of the Week below), is a bit of a ‘show-off move’; he made sure to place the numbers 15 and 14 side by side on the bottom middle to mark the year of his work! 

However, the patterns did not stop there. People discovered that a number of different patterns in his magic square all summed to 34. The main diagram shows a few of these patterns.  It seems Dürer was onto something. Further research discovered that there are over 80 different geometric sets of four that sum to 34! How many can you find?

Even today we are not sure why this is so.  Perhaps a link to the divine? The Chinese were certain God placed these patterns that they included magic squares in  religion.

Magic squares are not simply a numerical curiosity. At the end of the 18th century, the great mathematician Leonhard Euler started a mathematical examination of versions of magic squares called ‘Latin squares’. A Latin square is a square grid filled with symbols that occur exactly once on each row and column. The second image accompanying this article provides an example of a 4x4 Latin square.

To this day mathematicians continue to research Latin squares as a bridge between several branches of mathematics such as algebra and combinatorics. Latin squares have direct modern applications to cryptography and advanced telecommunications.

Dr Beatriz Zamora Aviles is a lecturer at the Department of Mathematics within the Faculty of Science of the University of Malta.

Sound bites

• In 2018, a non-mathematician solved part of a difficult mathematical problem that had remained open for over sixty years. The Hadwiger-Nelson problem involves a collection of points connected by lines on a plane. All the lines are exactly the same length and every point is given a colour. The problem is to determine the minimum number of colours needed such that all directly connected points have different colours. Previous to this breakthrough, it was known that this number is at least four and at most seven. Aubrey de Grey enjoyed solving puzzles in his spare time. He loved Othello and played regularly with a group of mathematicians. They mentioned the problem to him and he started working on it. The ‘Moser Spindle’ is a small arrangement of points as described in the problem that needs at least four unique colours. de Grey combined a number of Moser Spindles with another collection of points to form a 20,425-point monstrosity that needed five colours. A breakthrough! He then improved his solution by using approximately 1500 points and submitted it to the mathematical community. His paper, entitled ‘The chromatic number of the plane is at least 5’, is published in the mathematical journal Geombinatorics (Year 2018, Volume 28, pages 18–31) and is freely available at https://arxiv.org/pdf/1804.02385.pdf.

For more sound bites, listen to Radio Mocha: Mondays at 7pm on Radju Malta and Thursdays at 4pm on Radju Malta 2 (https://www.fb.com/RadioMochaMalta).

Did you know?

• 111 is the magical constant for a 3x3 magic square composed only of 1 and prime numbers that are not necessarily consecutive.

• Can you construct the magic square? The numbers you have to use are 1, 7, 13, 21, 37, 43, 61, 67 and 73.

• The opposite sides of a die always add up to seven.

• Benjamin Franklin, one of the founding fathers of the United States, invented a 16x16 magic square. Although the diagonals of this square do not sum to the magical constant of 2056, there are many other magical properties. Benjamin Franklin was quoted as saying it “is the most magically magical of any magic square ever made by any magician.”

For more trivia, see: www.um.edu.mt/think