A magic square is a combination of numbers arranged in a square such that the sum of each row, column and both the corner diagonals is the same. This sum is termed the summation (S) of any given magic square.
Magic squares have existed since prehistoric times. Examples of magic squares have been found in ancient Chinese and Indian literature. As early as the eighth century, Arabian astrologers extensively utilized magic squares in astronomy, using magic squares to construct horoscopes. It is likely that magic squares were so called because the properties they possessed seemed to be otherworldly, extraordinary and wonderful. German scholar Cornelius Agrippa even devised magic squares of orders between 3 and 9 and associated them with the celestial bodies Mercury, Venus, Mars, Saturn, Jupiter, the Sun and the Moon. In fact, magic squares were so superstitiously revered that many people engraved them on silver vessels and considered these vessels as “good luck charms” against the plague. Although magic squares lack any immediate practical applications, their position in the world of recreational mathematics has been evergreen.
Characteristics of Magic Squares
There are many different types of magic squares. Magic squares of odd and even orders are governed by different rules of construction, so we’ll study them separately. This is obvious, in that the starting step in the construction of odd magic squares usually involves the center box. Even magic squares do not have center boxes, so we employ a different tactic.
There exist 880 squares of order four, manually counted by Frenicle de Bessy in 1693. The number of 5×5 magic squares was similarly found to be 275305224. And the number of 6×6 squares is not known, because there is no clear-cut method to estimate the number of magic squares of any arbitrary order n. However, scientists have used Monte Carlo simulation methods (intensive random sampling of magic squares) to approximate the number of 6×6 squares to 17745000160000000000.
There exists a magic sum, or magic number, defined above as the summation S of each magic square. For a magic square of order n, the magic sum S is
Magic Squares of Odd Orders
A magic square of order 1, with the single entry being 1, is obviously a magic square. Let us consider the next smallest odd magic square, the 3×3 square. There exists only one magic square of order 3 and it has been known since pre- historic times. It was so revered that prehistoric Chinese mathematicians even had a special name for it- Lo Shu.
Knowing all the properties of magic squares, let’s look at some methods to construct them.
The simplest method to construct odd order magic squares is the de la Loubere method or the Siamese method. We start by placing 1 in the central box of the top row. Now we can fill the boxes in increments. Place following numbers 2, 3, 4... etc in the square to the top right of the previous square. If no such square exists, we follow a ”wrapping around” method. If no square exists to the top or to the right, we respectively return to the bottom and to the left, in that order of priority. If this square (s) is/are filled we place the number below the previous one.
Illustrating on a 7×7 square,
Can you figure out another method to construct this square? (Hint: Look for patterns in the positions of the odd and even numbers and the shape their boxes make)
Now how do we construct even order magic squares?
This is not a cliffhanger! Check out my article on magic squares here for methods to construct even order magic squares, brain-teasers, exercises and more!
Bonus question: Is it possible to construct a 6x6 magic square with the first 36 prime numbers? Prove your claim.
References
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