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What is a Magic Square?

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By Chandan Ghughtyal

I usually start my lessons in my classroom with an engaging warm-up or introductory activity that stimulates my students’ interest in mathematics. I recently asked in my D form to use the numbers 1 through 9 to form a magic square. While some of them found it difficult, others finished it in a matter of minutes. I saw several students get interested in magic squares as they worked. They inquired what a magic square is. Who thought of it? Why do we investigate it? And the classic, will it be on the May Test?

A magic square is a square arrangement of numbers where the sum of the numbers in each row, column, and diagonal is the same. This sum is known as the magic constant. Because of their mathematical elegance and symbolic meaning, magic squares have fascinated people for centuries.

Magic squares have an interesting origin. According to a Chinese tale dating back to around 2600 B.C., the first 3×3 magic square was discovered by Emperor Yu on the shell of a turtle. This discovery, known as the Lo Shu square, is believed to bring stability and prosperity.

In ancient China, during the reign of Emperor Yu, there was a belief in the Kingdom that offering sacrifices, particularly grains, to the river God, known as the River Lou, could prevent floods. Emperor Yu himself once went to offer sacrifices to the river. During this event, an unusual occurrence took place: a turtle appeared from the river, having a unique pattern on its back. The emperor and those around him were astonished to see this pattern, interpreting it as a divine sign. They carefully noted down the pattern and believed it was auspicious. Since that momentous event, the pattern has been regarded as a symbol of good fortune and prosperity. As it is thought to bestow prosperity and stability upon the occupants, it has become common in China to include this fortunate design at the foundation of all new construction.

Over time, the magic square spread from Asia to Africa, Europe, and other places. It became an important part of different cultures and was used in many ways, such as for beauty and fun. Despite once being considered magical, magic squares are now mostly valued as challenging math puzzles. Still, their transition from mythical beginnings to mathematical challenges shows how people have always been fascinated by patterns, symmetry, and the mysteries of the world.

How to make a 3 by 3 magic square? By taking any value of x the magic square can be formed as follows. By letting x as 10 a magic square is constructed in which the magic constant is 30.

To construct a magic square of 4 by 4 we can take any 16 consecutive numbers and arrange them in a 4 by 4 grid. Then by rearranging the certain numbers in a particular order will give a magic square.

By interchanging 2 and 15, 3 and 14, 5 and 12, and finally 8 and 9, the above arrangement will give a 4 by 4 magic square. The same can be done by interchanging the corner numbers diagonally and the numbers of the inner 2 by 2 grids also diagonally, giving a 4 by 4 magic square with a magic constant of 34.

When each number of a magic square is multiplied or divided by the same number, or when the same number is added to or subtracted from each entry, the magic constant also changes the same way.

Even with the date of birth of a person, we can create a magic square of 4 by 4 order. Let ‘D’ be the date, ‘M’ be the month, Y is the first two digits, and Z is the last two digits of the year. The date of birth of Krtin is November 20, 2006. For him, D is 20, M is 11, Y is 20, and Z is 6. By following the set arrangement in the grid below, the birthday magic square of Krtin can be constructed. The magic number for Kritin is 57.

In addition to being fun, magic squares encourage mathematical curiosity and problem-solving abilities by displaying number patterns and symmetry. They represent harmony and order in the cosmos and are present in many cultural practices around the world. Magic square research demonstrates the connection between mathematics and human expression, stimulating creativity in literature and design. Magic squares improve critical thinking, creativity, and a respect for differences in culture.

To create a 5 by 5 magic square, start by arranging numbers from 1 to 25 in square order. Then, draw six parallel lines slanting from right to left and six parallel lines slanting from left to right. Adjust the numbers that are left out of the grids, such as 16 and 22, moving to the 5th grid from left to right, and 4 and 10 moving from right to left. Similarly, 1, 2, 6, 24, 20, and 25 will shift to the blank grids. With these adjustments, a magic square with a magic constant of 65 is constructed.

Thus, making magic squares is a fascinating activity, irrespective of its historical, cultural, and aesthetic importance. To learn mathematics, it is essential to have a logical mindset. There are various methods and various types of magic squares; studying these patterns and arrangements will surely enable learners to develop an interest in the subject. The starter activities, like playing with numbers, stimulate the development of mathematical thinking.

(Chandan Ghughtyal is a Mathematics Master at The Doon School)